Chevy, I'd definitely recommend Dana, either as an acclimatization hike for a fourteener or a goal unto itself. It's probably the shortest/quickest way to get to 13K' in the Sierra, and the payoff is fabulous. Once you reach the crest, though, the wind can be wicked. My wife and 11 YO daughter did this with me a few years ago prior to a Whitney backpack, and we were up and down in what I remember as a relatively short, leisurely day. I can't recall the exact duration, but it was a late start and early finish, with lots of goofing-around time in the snow with our daughter and maybe an hour on the summit. I do remember that we got to the Whoa Nellie Deli before the dinner crowd hit.
I also think Little Lakes Valley is an excellent choice as the first hike of the trip for a family. Gorgeous scenery and not very demanding. Great way to start acclimating without pushing it.
I know you have your accommodations in Mammoth already set but in the future you might want to consider renting a condo. The inventory is heavily glutted there in the summer and they can be dirt-cheap. Certainly the best way I've found to sleep high in comparative luxury in the Sierra.
