I'm actively involved in alternate history discussions on the alternatehistory.com website, so I think I may have a special view on this topic.
"Assignment: Earth", were the back door pilot still made and still the exact same thing in this world where Star Trek was somehow a big success, would be the obvious spin off possibility. It would also have the strongest chance of any listed. Like a lot of shows of the era, it was related to what it was spun from, but properly not the same thing so I do get a vibe that it would work.
The ideas some people are proposing about a show on a different starship are ideas I disagree with. That thought process comes from the TNG era, where that show spun off Voyager and Deep Space Nine and Enterprise. I do not believe the same thing would have happened had Star Trek (TOS) been successful. Think to what actually happened when TNG was being talked about and went on air: people were saying how could a show without Kirk and Spock work and nearly everyone expected it to fail. In this alternate universe, Star Trek is a success, putting even more pressure on not doing a different space ship since the thought of "how can you do it without the Enterprise and Kirk and crew" will be even stronger in that case.
The most possible of those is a space station based show, and I also disagree with that because the same criticism will be with that show as was aimed at DS9 before it went ahead: how are you going to have adventures when the space station doesn't go anywhere? For those of you who will say "character development and complexity" or something to that effect, this is the 60s. Not only that, but this is the 60s with Gene Roddenberry's Star Trek. Roddenberry was against strong character changes, story arcs, etc even in an era when they were not especially strong (story arcs especially), because he wanted people to be able to watch any episode out of order and have it still work. Roddenberry and the era won't let a space station show work.
The most likely thing for a spin off would just be a follow up series at some point. Just to have Shatner and the rest of the cast, with some new people or replacements here and there, brought back sometime in the 1970s and put aboard the Enterprise or a refit Enterprise or an Enterprise II or whatever it would be.
If you're interested in this topic, I highly (highly) recommend an ongoing alternate history timeline called "That Wacky Redhead". It's about Lucille Ball deciding not to sell Desilu to Paramount, and thus she cheerleads Star Trek and gives the show the chance no one else did and it becomes a success. One of the things covered is a spin off (I can't remember if "Assignment: Earth" is made into a spin off in the story, though). It covers not just Star Trek, but the various other changes to history that come from that initial point of change, such as Laugh-In being canceled, a different outcome in the 1968 election, and Shatner's ego inflating even more.
You can find it here:
And a simple chapter list (to avoid jibbah jabbah) here: