Why are there so many top ranked answers for how to get the dye out of your or your kids' hands that just don't help? Dishsoap? Baking soda or toothpaste plus lots of scrubbing? Feh. Well, consider that baking soda does provide great cleaning power and abasion, but vinegar is what they use to disolve the dye packs in the first place. I figured, why not use both?
I put a small pile of baking soda into my daughter's hand then sprayed some vinegar on it (I had some in a spray bottle, but you can just pour it). It started bubbling as they do when combined and I helped her rub it semi-vigerously over her hand for a few seconds. A good rinse and pop; there you go!