My grandfather's doorknob and lock are rusting...help!

what material is it? copper, iron etc.

Try Brasso to clean it up

If it is brass, KETCHUP works wonders! Just put on paper towel and rub! It really works!

I am not sure what the metal is, but you might want to consider using rust converter. Here is a link to see what rust converter is: http://amzn.to/2kTcYBb

Handan

Looks to me like the finish has completely worn off in certain areas - that will never buff back.

I suggest you begin by removing both the handle and the lock, use steel wool to buff it a bit, clean and then spray paint with some of those nice Rustoleum paints in a nice brass, or even a steel color would look nice. I think you might even find an antique bronze color - that would look swell. When painting the lock, I would suggest you put in an old key so as not to get paint inside the keyhole.

Then reassemble. If the lock is not working well, then spritz a little graphite power in and turn the key back and forth a few times. To paint the lock, I would suggest you put in an old key so as not to get paint inside the keyhole. All this is extra work, but it will be worth it.

PS: I buy my Rustoleum paints at the Habitat Restore for $2-3. Wish I could give you one of mine!