destination for one day between Cambridge and London

Cambridge and London are too close for there to be an obvious overnight in between. It is only an hour by train and some people commute.
Peterborough has a cathedral and a train service from Cambridge and is worth a few hours. Ely is nice and a short train ride away. But neither are worth an overnight stay. Lincoln is as you say 2 hours in the wrong direction.

Oxford is a good place to "compare and contrast" with Cambridge, but thre is no direct train route.

There are lots of historical locations in the are, but they all need a car.
If you had a car I would suggest half a day in Ely and half a day in Wimploe: http://www.nationaltrust.org.uk/wimpole-estate

With only one night, I would go to Ely. We found it to be a very pleasant town, with a nice cathedral (you have several tour options, depending on how in depth you want to go -- look at the cathedral website to see what is happening on the day you'll be there). We also enjoyed Oliver Cromwell's house which is close to the cathedral. We also visited the cathedral in Peterborough. Peterborough gets fewer tourists, so they don't have as many cathedral tours as Ely. Again, check the cathedral website to see what is available. I would choose Ely between the two.

We loved Lincoln, another great cathedral, with some interesting tours. Also the castle (you can walk the walks and tour the Victorian prison) and the new Manga Carta vault. Check website to see if the Magna Carta will be in town when you are there. There are only 4 surviving copies of the Magna Carta so it is cool to see. We spent 2 nights and had 1-1/2 days to tour and didn't run out of things to do.

Stamford is a Collyweston stone town easily reachable by train.

I can give you a few suggestions of lovely places to visit relatively near Cambridge. Most would be daytrips but you could choose to stay overnight in either Ely or Bury St. Edmunds. We used to live nearby and I have visited and enjoyed all of these places. If you have any interest in aviation and World War II visit the Imperial War Air Museum at Duxford, very close to Cambridge. Also not too far from Cambridge is the Wimpole Estate, a fine old country manor. There you can explore the farm property and manor house. Right on the edge of Cambridge is Anglesey Abbey. The gardens and path alongside the river leading up to the old mill are beautiful and very photogenic. I second the recommendation to visit Ely. Ely is a great small town for exploring and it is close to Cambridge. Its imposing cathedral is visible for miles around. The town is small and easily walkable. You can easily take a train or bus from Cambridge there. If you go, make sure you go down and walk around the riverfront, not far from the train station. A little beyond Cambridge is the town of Bury St. Edmunds in Suffolk County. This town is also accessible by train. Beautiful gardens at the abbey ruins and colorful flowers adorn the entire town.

Thanks for all the suggestions thus far! I think our Cambridge friends may take us to Ely or Wimpole while we're visiting, but I'll certainly check out those other possibilities. My latest brainstorm for this extra day in question was to simply head toward London and go right on out to Windsor, stay there the night, and do Windsor castle the next day before heading into London for our 4 nights. I found a couple Air BnB places right near the castle. That way, we wouldn't have to go out to Windsor on a day trip while actually staying in London--we could focus on the proms and other things in the city.

now that may be the best yet

I think that Windsor plan makes sense too. Or Ely if your friends don't take you there.