First question: How old is your daughter?
You would also want to consider your daughter's temperament, but I'm guessing she is fairly young if you would be making the decision for her. For arts that would prevent injury which also has health benefits, you're looking at internal arts for the most part. Aikido and Tai Chi fall into that category. There are some external styles like certain types of kung-fu which is also healthier like some have already pointed out, but what it comes down to is whether the teacher has an understanding of the internal principles. It would be a bit of a stretch for kids to get into the internal arts anyway, but there are definitely additional benefits especially if you do. Internal arts also tend to take longer to learn in terms of being proficient in self defense application. Whatever you decide, consider the quality of the teacher. There are many McDojo's out there, and it's questionable whether you learn anything useful at all from the typical generic school.
BTW, for those with an interest I wouldn't mind taking Bagua at some point in spite of the fact that I am no longer interested in the martial arts in general. I am particularly interested in the relationship between the art and the I-Ching, (wooo, deminz!) and I understand it is good for fending off multiple attackers in terms of the combat value.
I've looked into Aikido online. it sounds very good but it also seems that there can be a quasi-religious or heavy life-philosophy aspect to it that makes me a little leery of it. (Don't want to be going from a religious cult into a martial arts cult after all.)
Yeah, let me tell you about that aspect - its probably either because the people who try to do it out of respect don't actually get it, or it is your perception from having been a JW. I remember when I first tried Tai Chi, I was so anxious about having to bow to pictures because of the whole idolatry thing, but it's really just a matter of respect. Obviously any reluctance to do so is just because of the way you were taught. I wouldn't hesitate to bow to any of you, it doesn't mean you are worshipping anyone or even that they are better than you as a person. A lot depends on how you hold it. As for the whole overly polite propriety business, I've found that is simply the confusion of well meaning students. Anyone who actually gets it would know that bowing in itself doesn't mean anything, and you can't force anyone to show respect - at the most they would just show you the ritual and tell you why they do it. You always have the option of just not taking it so seriously like they do, that is not getting overly caught up in it.
Specifically regarding not wanting to get into some kind of cult, remember it takes two - you gotta buy into it. Being afraid there may be some cultish element would be like the whole fear of demon thing, if nothing else the demon is in your own mind.
The fact is even those without a 'heavy life-philosophy aspect' will have something to say along those lines, it would probably just sound like Tony Robbins Personal Power instead of something more traditional. If nothing else they might just have a little code of conduct like Mr. Miagi's "karate for defense only" deal. The interesting thing is the ones I know that has the least to say in this regard and just kind of seamlessly teach without a lot of artifact in this way are actually some of the most spiritual people I know with a specific path, but that's the Taoist way - they don't tend to get preachy and haven't much use for ceremony.
