Does it eventually go into a sleep mode when the screen goes blank? If so, it should be OK to leave on, and waking from sleep is probably easier on it than starting up from scratch.

If you decide to leave it on and it doesn't go to sleep, I'd do everything possible to provide as much air circulation as possible to reduce overheating - make sure there's lots of room around any air vents, keep it out of sunlight and in the coolest room, etc.

If there's a way you can do it, I agree with Giok that taking it somewhere for an assessment and recommendation is a good investment. And backing up the files was a great first step! (Did you get her e-mail and address book? They're often hidden is quirky little places.)

Good Luck!
JudyB