You can skip the backstory if you just want to get to the rings...
Backstory: I used to answer phones for a living, and most of the time I didn't have to use my hands for anything terribly important while talking to people. So, to keep my hands occupied, I decided one day to bend a paperclip on my desk into something a little more interesting than a paper clip. The end result was that I made several small wire sculptures (mostly little fantasy style characters) with the remains of the box of paperclips, and decided that I should buy some craft wire. I began making all kinds of simple little things from wire and brought some little wire flowers home for my wife and daughter. My daughter now thinks I can make just about anything from craft wire. Long story short, I have some craft wire that I keep handy.
Now on to making the rings...
Step 1: Materials
Craft Pliers (or simple needle-nose will do, but you may end up with some teeth marks in the craft wire).
Step 2: Straighten and Then Wrap the Wire
Once the wire is sufficiently un-kinked, begin wrapping the wire to the appropriate size. I used my first knuckle on my pinky because that gave me a ring that would fit my daughter's ring finger. Wrap the wire around several times at the desired diameter. The number of times will ultimately depend on how wide the string loop is on the back of the button. Too much is better than too little for this step, since you can always cut the wire shorter, but you can't make it longer without starting over. Cut the wire once you have enough wrappings (This is really just a best-guess, so again, cut long just in case.)