That "higher summits forecast" is one of the most accurate weather prediction models on earth. They are far more accurate than weather.com, noaa.gov, etc. They update at 5 or 6am each morning.
The easiest route up Mt. Washington is the Ammonosuc Ravine/Jewell Loop (10 mile loop). It's extremely scenic and features waterfalls and a stop at an AMC hut (where you can rest, refill water, buy snacks, or even stay over for the night ($$$$)). Any athletic kid is going to love this loop. It's also less exposed than Tuckerman's Ravine.
I suggest packing gear sufficient for 40 F and pretty strong winds. These were the conditions at the summit in late August a couple of years ago on the day I climbed this mountain. (It was 65 F and breezy down in Pinkham Notch where I started.) I had a couple of layers underneath a good windbreaker, a headband to protect my ears, and gloves - and hiking poles to help me stay upright!
There should be no problem for an athletic 12-year-old to climb it on any major trail. As always, I recommend taking a good map - I used the AMC's White Mountains set.
Thanks for all the tips. Washington provided an ample glimpse of how bad it can be. We did the ammonusac and it started out muggy. As we rose the rain turned from mist to hard drops. The temps were dropping as well. We were soaked when we got to the hut and visibility was nil. It was nice location, we actually could not see the real hut when we got to the waste but below it. We changed to dry raingear and headed to the top. Visibity was poor and the cairns were nice. Quite frankly we both had a blast. She suggested we get Mansfield the next day so I love that she got the bug. I had to explain that most mountains do not have huts, especially as nice as that one. Definitely deserving of a nice donation!
I am glad I packed all the clothes that I did, we used them. So yes being prepared does not go out of style. The waterfalls were beautiful, just wish we could of seen more than 50 yards at the top.