I am glad to see that I am not the only person to have thought about this.
My first thought was to use some thing like a pistol light modified to mount on the rail that holds the sight, but I think it would cast a big shadow.
Another was a Twofish Bikeblock on the stabilizer, but it would hold the light parallel to the stabilizer which might not be any good, and I wouldn't want any thing but a small light on the stabilizer either.
I like the idea of using the stabilizer mounting hole but it isn't any good if you plan to use the stabilizer.
I think it would probably be best to keep the weight as close to the riser as possible. At this point I am thinking zip-ties vlecro and some small pieces of rubber to keep things form sliding around.
I don't know why I think about this so much I don't shoot any where that doesn't have lights.
You didn't mention if you had a traditional recurve or a modern recurve. If it's modern then it's probably threaded for a stabilizer. All you need is a 25mm to 25mm or 30mm to 30mm flashlight mount. Look for shotgun barrel to 1 inch mounts too. Then you'll be able to mount any 1 inch or 30mm flashlight....many to choose from. Whatever it is, it will have to be bright because the amount of light coming through the peep aperture will be small and sometimes it's difficult to find with your eye. Once you do find it, it's difficult to see your pins unless there is a whole bunch on light on your target. I'd say 120 lumens minimum or you're handicapping yourself. In order to see your pins easier attach a little red or green LED button cell light to your sight bracket. Velcro tape will work.
If you have a traditional recurve........good luck on mounting anything on it. Another problem that you'll run into is being able to see your arrow in order to aim (if traditional).
If it's traditional I would suggest a headlamp. You may be able to position it to shine on your target while having a bit of spill light to see your arrow.
Good deal the modern bow gives you some options. I'd recommend on line shopping for a specialty item like that. Here is the basic concept of the "tube to tube" mount. http://www.botachtactical.com/msshtrbamo.html
That's one is kinda beefy with the rails and all. But something like that should do. Another option is "tube to rail" in this case you'd have a tube clamped around your stabilizer and it would support a built in weaver or picatinny rail which would allow you to mount a weapon light. More expensive because it limits you to weapon lights only. Also, the shorter the stabilizer the better. You'll have to attach the light near the end of that stab. in order to avoid a huge shadow. Best if it's close to the riser and your hand