Without seeing it, we would be guessing like you all the way to the store.

 

Buy the washer assortment. That way, you might have what you need.

 

You have the general idea, so know which tools to take with you, but, as Larry said, you're unfortunately not going to know for sure until you get there. Having done projects like that myself though, repairing items a distance from home with not a lot to go on, I would get online and jot down the addresses of a few hardware and/or big box stores.

 

I would not buy anything until I got there.
As as old as that likely is your going to be really lucky if the shut off valve even works so you may also need one of those.
If your really lucky and it is just a simple washer the seat may also need replacing so you would need a seat and the tool to remove it.

 

If the handle rotates 90*, you may have a washerless stem. You will need to turn the water to the faucet off, disassemble the handle , remove the stem, go to a plumbing store to find a matching stem.

 

When u do repair her faucets, show her/demo how it’s done so she (or other members of her household) can do it herself next time. It’s not that practical to depend on someone who lives 50 miles away to come over to repair faucets.

 

Time is money, and if your time is valuable, you may want to consider just replacing the faucet, rather than potentially making several trips to the store (or several stores) hoping they have replacement parts for that particular faucet.