Yes, spend a night in Hakone. I second what others have said. But, for whatever it's worth, I'll also point out that it's also doable as a day trip (or en route to Kyoto). I will list some things for you to consider...

- Doing Tokyo to Hakone, then continuing onto Kyoto on the same day is probably only feasible if you don't have a large bag with you. It's hard to hop around Hakone with a bunch of belongings. But if you just have relatively small things, then you can stash them in a locker at Odawara Sta or even Hakone Yumoto Sta. However, large suitcases can be tricky. While Odawara Sta has some large lockers, large station lockers often are sold out and so you can't count on them being available. So if you have a large suitcase and want to do a day trip to Hakone, you'll have to leave it at the hotel. If you don't have a large suitcase, then continuing onto Kyoto from Odawara via shinkansen Hikari is only 1hr more than returning to Shinjuku from Odawara via Romance Car.

- Keep in mind that the Romance Car only goes to Hakone Yumoto, which is essentially the base of Hakone. A trip to Hakone should at least include taking the Ropeway up to Owakudani, and it's 70-90 minutes to get from Yumoto to Owakudani via combo of tozan densha / cable car / ropeway.

- An example day trip could include: 2hr at Chokoku no mori; 1hr at Owakudani (take Ropeway up); 2hr stop in Yumoto for a soak in a day onsen plus some shopping. Including all the transportation time, you could fit all that into a 9 hour time frame from the time you arrive at Yumoto until the time you depart from Yumoto. But if you're going to do all this, then I'd suggest arriving at Yumoto earlier than what you have planned right now. There is a Hikari leaving Odawara at 6.08p (arrives Kyoto at 8.11p).

- But again, it's best to spend a night and enjoy the onsen hotel.