I found two different ideas - both great and very easy. The first idea is the simplest - turn your tights into "fingerless" gloves by simply cutting off the legs and toes and adding a small hole in the heel for your thumb to go through. I absolutely love the second idea I found - turn your tights in leggings. This took a little longer (about half an hour with my sewing machine, since I'm a little out of practice), but they turned out great (forgive me for the poor lighting in my photos - it's really the same pair of tights, despite the color differences).
Annika loved my new creations. The fingerless gloves are a little big on her (understandably, since her arms are smaller than her legs), but they've already been referred to as "princess gloves" and a "cast" for a broken arm. I'm thinking they'll get lots of play as dress up items. The leggings are great, and I can see her getting a lot of wear out of them. I'll probably do this with all her other tights, as she wears them out. Even the ones that are too short for full-length leggings would make good capri leggings for those "in-between" type days (which sadly it looks like we are past - brrr it's chilly today!).
Hurray for upcycling!