Definitely experience soaking into onsen! Hakone is very famous as an onsen resort. You should not miss onsen and staying in an onsen ryokan.
There are many museums in Hakone. Such as glass museum, open-air museum with lots of sculpture pieces, Little Prince museum, etc. People like to visit around museums.
Lake Ashi is also a famous attraction. You can get on a boat cruise with a fancy boat. The view is quite nice and you can see Mt. Fuji if the weather is nice.
Near Lake Ashi, there is Hakone Shrine. The huge torii gate will surely amaze you.
There is also cable car and ropeway. Owakudani is very famous as an active volcano. Black boiled egg sold there is very famous!
And there is Gotenba Outlet Mall near Hakone. You can shop until you drop!