Before Sony morphed a couple of traditional point-and-shoots into cans that you clip to your smartphone, Canon's little square PowerShot N was already giving people a pocketable, attractive, and fun option for snapping quick photos and movies and sharing them on the go. If mom is looking for something to supplement her smartphone shots, this is a good choice.
Part camera, part Android mobile device, the Galaxy Camera 2 gives moms a way to shoot photos and movie clips and edit and share them with whatever app or service she wants. And if she wants to send an e-mail, stream some music or video, or read her favorite Web sites, she can do those things, too. No need for Android? Check out the Samsung WB350F Smart Camera, which packs a lot of the same features with a traditional camera interface and for less money.
If your mom has been complaining about missing shots because the subject is just too far away, the L820 and its 30x zoom should do the trick. It's a 2013 model that's still around at good prices, but if you can't find it, consider the L830, which is basically the same camera but with a 34x zoom and a tilting LCD. It's geared for automatic shooting, so if your mom is the controlling type, check out the Canon PowerShot SX510 HS instead.
A compromise between mom-deserves-the-best prices and kids-can-afford budget (at least if they all chip in), the RX100 was our top compact pick until unseated by its subsequent, more expensive sibling, the RX100 II. It's comfortably small and delivers excellent performance and photo quality.