My mom asked me to find her some new games. Need to be simple, score based (so it needs to record scores) and can't be too long to beat a level, something she can play for 10-20 min on one sitting.
Free would be advantageous.

any suggestions?

Bejeweled (I prefer #3, but any will do it). My mom loved it... she switched to Catan now, but still enjoys Bejeweled from time to time

you have to buy it right?

tetris
That might have been too obvious but Ubivis stole my answer

yes, but it is damn cheap :)

Shatter. It was around $2 on Steam the other day, it fits all your criteria, and it's great :)

Chuzzle. I usually hate matching games, but I liked this one :)

I quite liked Fractal, but it's $6.99 and takes some time to get into, even though the idea is simple (a bit abstract and no real tutorial, once you figure out how it works it's fun though). It records only your best score per level and without name, but it provides satisfying mental exercise for a few minutes at a time.

Puzzle Quest is something you can do in 10-20 min bursts, can't recall if it had scores per se, but could use it as a gateway to get her into other way.

In terms of match 3 games, my mother and aunt both absolutely love the Fishdom games. They are pretty relaxed and chillout games, and very easy to play in small bursts. Very recommended for match 3 stuff.

You can grab them at Big Fish Games, and while not free, I think they are running a $3 promo for new users. Really can't beat that.

Super crate box is free.

If you have any humble keys you may already own Jamestown. That can be simple on the lower difficultys and levels take a few minutes.

Come to think of it, a lot of the humble bundle games fall into the category of easy to pick and play. I guess if you do have any keys lying around just check. The latest bundle has Toki Toki. If you don't feel like spending money on it you can pay whatever you want.

What about Majhong? Some games will maybe take longer then 20 minutes, but she can save the game and go back to it later. Also there is a highscoreboard (playing against time). And the best: It's free.

http://opengamestudio.org/lang/en/ogs-mahjong

Con: It's 3D and so older graphic cards can be a problem.

Pinball games? Gog has a nice selection of them

Peggle, a no-brainer.

If it has to be free, then Peggle Deluxe (if I remember the name correctly; anyway, the Half-Life one) though it has pictures of scary headcrabs IIRC.

If you're not a Steam hater, Plants Vs Zombies is still silly cheap for the next few hours and could fit the bill.