The major cooking advice is to buy a certain cut (sirloin, etc..) and cook it to rare-degrees.
However, my guests don't like beef that has any pink inside or "blood" (I know it isn't blood) oozing out. I don't assume the same cuts still produce the tastiest steak?
What are the properties of raw beef that result in a great well-done steak and which cuts normally present these properties?
I'm rather surprised by the judgmental tone in some of the answers here. A well-done steak is a culinary preferência; just because you don't share that preference is no reason to be rude about it. Some people like caviar; others don't, despite the fact that it is expensive and lauded by many "people in the know." Some people appreciate an espresso made lovingly with freshly ground coffee in the "right" kind of grinder; others find it too strong or bitter and would prefer an "American style" of coffee with cream and sugar. Taste is subjective.
I'll admit something personal -- for the first quarter-century or so of my life, I only ate well-done or medium-well steak. It's how my father always cooked it at home on the grill. He didn't actually prefer steak well-done: when we'd go out to the restaurants, he'd generally order medium-rare. But even though he'd time steaks, adjust heat on the grill, etc., the vast majority of the time, they were well-done. I was used to it. I liked it, because it was what I knew. The few times I encountered less-done meat, I found the texture odd or even slightly off-putting.
Then, at some point, I was convinced to try more rare steak, and I soon accepted it. I now almost always order steaks medium-rare, and I do prefer them that way. But I also spent a lot of time ordering steaks more well-done (at restaurants, I'd almost always order medium-well), and to those of you who claim you can't tell the difference when a steak is that done, you don't know what you're talking about, because it's not your common way of eating and perhaps you've never had a well-done steak that was prepared in a reasonable fashion.
Anyhow, to answer the question: as some others have hinted at, choose a cut that has a "looser" texture with fat running through it if possible. Also, consider using a cut that you'd often tend to slice thinly when serving anyway, like skirt, flank, bavette, etc. You can also use more expensive somewhat fatty cuts (like ribeye), though the meat will toughen, so you won't get the benefit of the tenderness in such expensive cuts. Plus some of the cheaper cuts (or at least less expensive) have superior flavor.
Quality is actually amoras important in cooking well-done steaks, because older or worse quality meat with more connective tissue will become even chewier and tough when cooked longer. A somewhat high-quality ribeye cooked well-done can be a somewhat chewy but very pleasant caramelized experience with melt-in-your-mouth browned crispy fat interspersed. A poor-quality steak with poor-quality fat will just become tough and have its bad qualities exaggerated.
What you want to avoid -- unless your guests insist on them -- are lean cuts, which will end up tough AND dry. Filet mignon is a very poor choice (which will end up tasteless and tough), as is sirloin, as would be other lean tough cuts (like round). Also avoid cuts with a lot of connective tissue (but sometimes are sold as steaks to be cooked fast and rare), like chuck.
Marinating will help if you allow enough time for brine to soak in a bit (thereby adding not only flavor, but more moisture).
As for cooking, keep in mind many types of meat are cooked to "well-done" temperatures and still can remain juicy with proper technique (e.g., chicken). There's absolutely no reason to serve a tough, dry well-done steak unless you're incompetent. A "loose" textured steak as mentioned above that's sliced thin before serving will be chewy but won't necessarily seem "tough" if marinated and cooked properly.
How to cook properly? Do NOT do what most people do when cooking steak and just flip once. You'll end up drying out both sides of the steak by the time cooking is finished. Butterflying (which restaurants will sometimes offer to do for you when you request a medium-well or well-done steak) can be counterproductive for some cuts and can also dry things out more. (Restaurants do it mostly for their own convenience; it speeds cooking.) Unless you have a very loose-textured steak, which might benefit from additional browning reactions with greater surface area while not getting tough, you probably don't want to decrease thickness deliberately. Instead, keep moisture in with a somewhat thicker cut.
And flip often during cooking. It's more work because cooking to well-done takes longer, but it's the best way to keep juices moving around inside rather than boiling out the top while the bottom gets dry and burnt. (Think of what a rotisserie does; you're doing the same by flipping steak often.) Flipping frequently also can help soften fat and begin to break it down, which can add flavor and a "moist" aspect (if done right with high-quality meat, the fat might even be almost "melt-in-your-mouth"). Frequent flipping also aids in more browning reactions, which develop more flavor, and well-done steak does at least get that advantage of extra browning flavor (perhaps even crisped brown outer layers of fat). Obviously control heat; overall you'll need to cook at a slightly lower heat to avoid burning the outside before interior is well-done.
Then pull steak off while slightly less than well-done, and let it rest to creep up to well-done.
Para resumir:
I've accidentally overcooked skirt steak this way a couple times, and it was just as tasty (if not more so), basically as juicy, and almost as tender as if I cooked it to medium rare. (Skirt is always chewy anyway; well-done steaks will always be chewy, but they don't have to be excessively tough.)
Cooking steak to medium-rare and getting passable results is relatively easy with a thermometer. Cooking a decent well-done steak takes a lot more skill.
EDIT: A couple comments have noted that there are better ways to cook steaks. I absolutely agree. The OP didn't ask about preparation technique, so I was assuming a somewhat "standard" cooking technique for steaks (grilling, pan-frying, broiling, etc.), which my advice applies to. Personally, I'd recommend things like reverse searing or finishing in the oven, etc. too for better results, but I wasn't trying to turn this answer into "how to cook a steak" in general.
I would avoid "steak", which will dry out and become tough when cooked well done, and cook a cut of meat that is meant to be braised or grilled low and slow. That will mean that the cut has enough fat and collagen to break down, become tender, and remain moist.
I would look for cuts which can stand marination, and marinate them. Onglet (Hanger), Bavette, and Flank steak come to mind. The type of cut I'm thinking of tends to have a ropier texture, and be more highly-flavored, pungent, than the fine cuts. Still give them a high heat, to há pouco well-done or slightly under, and give a longer resting time in an only-just warm place, than you would for more tender cuts. Don't waste the juices, and slice thinly across the grain to serve.
It does not matter 'why' you want to make 'well done' steaks, if that is what you want to do, for yourself or your guests, it is 'ok'. No, it's not actually sangrento but some people are stuck with that perception in their mind, and it will not be erased by telling them "it's not really blood."
Ok, so you are going for two things "Steak" and "Well Done" here is what you are looking for: If you want "high quality" then look for a Flank or Skirt steak. These are thin enough that they will become well done before they surrender all of their moisture. Of course you are still trying to avoid 'the appearance' of less well done but with good marinade (say a fajita style) they can be 'moist' without the bloody/pink appearance. If you want to serve a more 'classic steak' (Rib Eye, T-Bone, Sirloin) you can do this, just don't waste the money of going for "Select" (or better) grades of beef. A well-done standard grade (off the grocer's shelf) premium cut is going to be every bit as good as a "Prime" cut once it is pushed to well done. (The quality of those better grades are more pronounced at a rare or medium-rare finish.
Regardless of what cut you choose, the technique that will render 'the best' results, however, is to slow roast the steaks (in your oven @ 275°F for 25-35 minutes) and the acabamento them on the grill or griddle to sear them no final. The slow roast will allow the steaks to achieve the desired internal temp (160°F) more slowly and then finishing them will put some maillard reaction on the surface to give a great flavor and juicy-but-not-bloody result.
They're the ones who are eating it; just give them what they want with a smile. It's no skin off your back. About 12% of Americans like well done steak, sometimes even with ketchup. And undercooked meat is a risk to certain populations with compromised/weak immune systems.
Use a thermometer to get it to well done, and no more past that (165 F); too many people blast their well done steaks well beyond this, giving well done steaks a worse rap than they deserve. If you're going to stick with cuts that come to mind when most people think steak (Ribeye, Strip, Tbone, etc.): Try to cook it as gently as possible as well; two methods I'd suggest are on wire rack in a baking sheet in a low oven (say, 250F) until it gets to around 160 F or sous vide. Then, finish in a ripping hot skillet with some oil and butter and herbs.
I'd suggest a cut like flank or skirt. These tend to cook quickly to something tasty and do fine well done (these are getting kinda pricy though). I'd grill or broil these cuts, and slice against the grain. The cuts that will do better at more done-ness are normally the tougher ones that have more fat and connective tissue (e.g. ribeyes do better at medium well than filet mignon; rarer isn't always better, you need some done-ness to deal with the fat and stuff).
Alternatively, if your guests want well done meat, try doing something other than a steak if it bothers you so much. Fajitas, for example. Or do a sauce, like this mushroom sauce. Or don't invite them.
Others have pointed out that well done steak is usually terribly tasteless shoe leather so choose the cheapest.
I don’t think anyone has mentioned the best cooking method if you really want well done steak.
Sous-vide it at 71 degrees C (160F) for an hour using something like esta, then fry it at very high heat for 30 seconds each side. You’ll obliterate any pinkness, and it’ll still taste vaguely like steak. The sous vide approach also pretty much eliminates the risk of getting more done than you want it - the internal temperature can’t go abound what you want - and it’s trivial to add any marinades or other flavourings you might want.
Can I suggest Korean style BBQ for an alternative to the traditional steak?
Korean BBQ usually use the same cuts of beef as steak, but are sliced thinner and often marinated. This allows them to cook well-done quickly, but don't loose as much liquid as a thick slab of beef. They also have added benefit of maximizing the surface area for that caramelized goodness.
Meat proteins dry out when they are over cooked. They squeeze out all their internal liquids and become dry and tough. This happens to all cuts of meat. If a person doesn't like their meat to be the texture of meat then this is their only option.
As moscafj wrote, cooking tougher cuts long and slow will melt the connective tissue and result in tasty meat again- the meat fibers are still dried out but it isn't tough. Unfortunately, meat prepared this way does not look or cut like steak.
Using cuts with a lot of fat, and cutting the meat thinly and against the grain will result in a well-done steak that at least has a chance of being chewed but it will still be far from anything that could be called a "great steak"
People who want their steak well-done don't want their steak to be meat-like.
The best cut to give them, therefore, is the cheapest so it will be less of a waste.
Incidentally- without exception, all the people who have told me they prefer steak well-done don't prefer steak. Given a choice they would have chicken.
Use chuck, or any cut typically ground/minced
Because if your guests are going to insist on well-done then you should be serving them burgers. It's one of the few ways to get well-done beef on a grill that's actually pleasant to eat, as the grinding process creates a nice, uniform distribution of fat and you can mix in other ingredients to help the burger retain both moisture and flavor. That counteracts the negative effects of cooking the meat to a higher internal temperature. And it's not something you can accomplish with a steak.
Just to be clear, I'm suggesting you grind the meat and make burgers; not that you buy a cut of beef that's typically ground and then try to cook it as a steak.