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It was implied that her illness was tuberculosis. In the novelized version of "Totoro" (illustrated by Miyazaki), it was stated that Mom was suffering from TB. Shichikokuyama Hospital, where she was staying, had a good reputation for treating TB, and that's why they moved there. The house they moved into was originally built a long time ago by a rich city man whose wife was suffering from TB. Granny used to work as a domestic for this lady, but the lady died. That's why the house is somewhat different from the other village houses, more Westernized, with gables.
This is a bit autobiographical: Miyazaki's mother suffered for many years from spinal TB, and she was away from home for a long time.
Mas de Wikipedia (spoiler):
The girls perch in a tree outside of the hospital, overhearing a conversation between their parents and discovering that she has been kept in hospital by a minor cold and is otherwise doing well.