I know this may sound crazy, but I'm hoping to see Kamakura and Hakone in one day - take the train to one city, then the other, then back to Tokyo. I plan to get up very early! Any advice?

pick one or the other.

Really? Do you think it's just not possible or do you think it will just be too much?

Which would you choose?

They are both really full day trips. When we went to Hakone we left early in the morning and did not get back to Tokyo until 8pm and we still felt rushed. If I were to go back, I would spend a night there.
I would pick Hakone if I could only see one because I absolutely fell in love with the open air sculpture museum there. I could have spent 4 hours there alone. Plus it's fun riding the train, the boat, the ropeway tramcar, etc. to make the circuit.

It is a bit much but it also is possible, assuming you are leaving from Tokyo or Yokohama.

Get a Hakone Free Pass and take the tourist route through Hakone. When you get back to Odawara station, you can get help at the info booth there for getting to Kamakura. Your visit to Kamakura would be to see the Daibutsu and not much else.

I did a similar trip. I started in Moto-Hakone and took the tourist route across Ashi-ko and the ropeway, etc. to Hakone-Yumoto then bus to Odawara. From there I used my JR Pass to Kamakura, back-tracked to visit an onsen at Atami and then went to Shinjuku for the night. Long day and it was a fair bit of travelling but the trains worked out incredibly well. When I did this I already knew my way around JR stations and trains. If you will be jet-lagged and on your first trip, then I would recommend taking glorialf's advice. I would choose Hakone.