Congratulations (to both mom and son).
Yes, I did almost two years in Japan. We were underway a lot (I was out of home port over 70% of the time I was stationed there) but there is plenty to do.
Public transportation is clean, safe and reliable (but rarely run all night, so make sure he plans ahead).
Yokosuka (pronounced Yo ko-s ka not Yoko sooka) is not that big of a town, but there are plenty to do there and all over the country depending on how much of the culture he wants to explore.
Close to the base there is Great Daibutsu (Buddha) within walking distance of the gate:
http://www.geckogo.com/Attraction/Japan ... at-Buddha/
Landmark Tower (also called Queen's Tower) is a short train ride up to Yokohama and has lots of western (US) style shopping and food. I strongly recommend Garlic Joe's and there is a Hard Rock cafe.
http://www.geckogo.com/Attraction/Japan ... k-Tower-1/
Just off-base in Yokosuka (two blocks on the left before Yokosuka train station) there is a hidden little Italian restaurant, Capriccios.
For day trips or weekend trips there are many schedule through the base MWR (Morale Welfare and Recreation) to include trips to pottery factories, climbing Mount Fuji (only open during July and August, another strong recommendation) and the like.
Mostly, I really enjoy just experiencing the culture, walking around and finding fun places to stop and eat and shop taking in as many experiences from a local perspective that I can, but again, that's just me.
There are lots of other shrines and temples all over the area and plenty of shops located near the base (and a decent exchange on base, also have him look for Takusan Treasures (gift and furniture shop) on base run by the Yokosuka area spouse's club).
Just a few ideas, I'm sure he'll have a great time if he can get away from base now and then.