The Hakone free pass (which isn't free) costs more on weekends than on weekdays. It's good for at least two days. If you buy it in Tokyo, it includes one round trip and is taken when you return, even if it would otherwise be valid for another day or two. There is a supplement for the wonderfully named "Romance Car" trains which are faster and nicer. I recall the price as being a bit more than the earlier poster noted, like Y4500-5500 each.

Since JR goes nowhere near Hakone, this is a good thing to do before or after your JR pass validity. Hotel rooms are apparently almost always available at Hakone Yumoto, but that town has no views of anything. Still, it might be a decent HQ town if you decide to overnight on short notice.

Edited to add: We quite enjoyed our Kiso "valley" hike and overnight stay but did the hike in the wrong direction and the uphill part was much longer and required much more vertical gain! I'd have to do some study to figure out the easier direction, let me know if this is in your plans.