I've done a fair bit of breaking (aka breakdancing), ballroom dance, and gymnastics, and a bit of rock-climbing. My opinions are as follows:
Gymnastics will develop all around athleticism, coordination, and flexibility. There are well defined progressions to develop your skills. Coaches are usually qualified and experienced. Training facilities are usually very nice (foam cell floors are great) with lots of toys. A strong focus on form is good for getting skills fundamentally sound, but I find the emphasis on trivial details (like pointing toes) annoying.
Breaking will get you pretty much what floor gymnastics does; it is basically applied tumbling. The coaching is usually less qualified, and the floors (sprung wood in most dance studios) hurt more when you crash. Form is not so emphasised. However you get a lot more freedom to create your own moves and follow your own path, and there are aspects, liking reacting to the music, that gymnastics doesn't have.
Ballroom dancing, at least to the level I do it, is fun but nowhere near as demanding as breaking or gymnastics.
While rock-climbing can make good use of existing flexibility I don't see that it develops flexibility. Balance will certainly improve, as will finger strength. It is quite expensive, at least around here, which is the main reason I don't do it more often. I don't think it's as balanced, in terms of all over body development, as breaking or gymnastics.
If I had to pick one purely from the standpoint of athletic development I'd choose gymnastics, because of the coaching and range of ways it gives you to assault your body (rings, floor, pommel horse, etc.) However I always recommend doing what you find fun, and there are enough gymnastic sports (add at least freerunning and capoeira to the above) that there is something for everyone.