Check this the morning before you hike: http://www.mountwashington.org/weather/ ... recast.php
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.
After the hike, go to the Upper Ammonoosuc Falls swimming hole (but beware in high water): http://www.newenglandwaterfalls.com/wat ... uc%20Falls
Eat at Moat Mountain (North Conway) or Flatbread (North Conway) or Red Fox (Jackson) post-hike for a great meal.