Each has its own merits:

Hakone: best for onsen ryokan, view of Mt. Fuji, beautiful scenery of the lakes, Hakone loop.

Nikko: UNSECO World Heritage sites so if you think old Japan with temples and shrines and like history, Nikko is the place. Take the 40 minute bus ride up to Lake Chuzenji as well for breathtaking views of the lake and nearby mountains and Kegon Waterfall, one of the few Japanese waterfalls I have gone to that truly deserves the hype. Onsen ryoken are available near Nikko as well although I found Kinugawa not as pretty as I thought.

Kamakura: This trip does not require an overnight stay. Rather, it's a true day trip. Visit the old temples and buddha statue, take the classic Enoshima rail, go to the beach to get a view of Mt. Fuji on a clear day, and stop by Yokohama at night for dinner and walk along the harbor. Kamakura may be combined with Hakone on the way back since they are in the same direction from Tokyo.

You can't go wrong with any one of these. My prefernce is in this order:

Nikko will be like nothing else you've seen, and is in a pretty setting with tall ancient trees. It's a long but relaxing train ride.

If you haven't already seen a bunch of temples, Kamakura is you chance to see nice old buildings in tranquil surroundings--start at Kitakamakura station. Skip the big shrine in Kamakura if you want to stick with tranquility. Easy to get to from Ginza.

Hakone is a nice excursion. Worth doing if you've seen everything else that Japan has to offer, but not (in my opinion) a must-see.

This question actually gets asked a lot in this forum and there will be many more opinions available to you if you search around a bit.

I don't like Hakone. Too touristy.

I like Kamakura because I like shopping & little trendy restaurants.

I like Nikko but it's a one-trick pony for me, ooh, look at the shrines, now let's go home. (okay, yeah, there's a waterfall too. I have NEVER seen the famous monkeys so don't even bring that up.)

Hakone is only worth the visit if the weathers good so that would be my last choice. We liked Kamakura but Nikko shouldn't be missed.

I actually do agree Nikko is all about shrines and nothing else so if you really like to see old Japanese architecture, shrines, etc, then Nikko's the place. Kegon Falls is very nice though, and Lake Chuzenji is truly magnificent, much more interesting than Nikko itself for me personally.

Otherwise, I actually did enjoy Kamakura a lot. You walk around in a relaxing pace, seeing whatever you like, enjoying restaurants and a bit of shopping.