From both player's and GM's point of view I've enjoyed most about games where players have lived in one specific location. They get familiar with the people and location, may start to like people there, get possessions or even rule a portion of the area.
This makes players attached to the area, and they'll have a motive to do something about things that happen there. It can be used to create dramatic events (a fire burns the largest city badly, killing many - and the fire was caused by a dragon. Now the fearsome enemy is even more fearsome and pc's now have grudge against it). Players get to know about local hermits, rumors and legends, odd locations, any of which can be used, and to which players can connect recent events.
The game should move from the base of operations from time to time. Various events may draw heroes to adjacent counties, and if they are forced to stay there, things may go badly in their country.
Primary source of inspiration for this topic
A great source of inspiration for building bases of operations for me (and my friends) has been computer games - especially Baldur's Gate II. Athkatla was in interesting location with full of interesting details - it was also visually impressive, which has also inspired my modelling. Most of the adventure focuses on the city, and the hero(es) always return to the town. And in the town they usually slept in Copper Coronet, an inn that is likely to change during the game.
But this isn't the only base. The hero may get a position of power and responsibility, from church, a noble's castle or other location, depending on his class(es). While these positions didn't feel that much like a base, as they added just minor tasks to the game, as an idea they are great. Combined to the base of operations -idea they pretty much form my idea of BoO -based games.
Another video game that has inspired me a lot is Knights of the Old Republic I. The Jedi academy in Dantooine serves as a kind of base, but only for part of the game. More importantly the small freighter, Ebon Hawk, acts as a base of operations. While having a ship as BoO may not feel that much home, it is one version of the concept.
The bigger the ship (whether space ship or sailing ship), the more it feels like a base. But to get a real BoO feel, I suggest adding a more stable base to the game. Having two bases may make this concept even better, and losing one of the bases still leaves another. And if you want to have two or even more bases, they may all be mobile. A city and a fortress (à la Edoras and Helm's deep) is a good example of having two bases.
Sometimes a base is an insecure location or vehicle, something that is the best you can have, but far from good situation. A ship is a good example of this, and even a large ship can be used very well this way - or what do you think of Galactica?
Insecure base doesn't have to be a ship though - it may be a small village where heroes have been stranded during the great zombie rise. They'd like to be somewhere else - and hope there still IS somewhere else - but the village is still a place they can call home. Sometimes safe haven can be used as a tool - something to make the players feel safe for a while, after which it can be snatched away.
Inspiration in TV series
Two of my favorite SCIFI series today are Babylon 5 and Galactica. Both of these have a large base of operations, first one having a static one (even if being a space station gives a certain taint of loneliness to it) and second one being a safe haven: A powerful battleship, but still without home base or allies and threatened by large hostile fleet. Babylon 5 is an excellent setup, both for the series and for roleplaying purposes.
In Fantasy, people travel around a lot more. There can be great places for bases, and heroes may even stay in those for a while or fight for them, but in the end they travel around a lot. And as traveling is slow, it takes a lot of showtime too. Two biggest examples that come to my mind now are Lord of the Rings and Game of Thrones. LotR especially is pretty much traditional travelling fantasy, but the world is well created and any games taking place in the Middle Earth can use bases of operations, and perhaps should unless they want to mimic adventures once seen. Minas Tirith and Rivendell seem like obvious locations for such bases, but I think that much more can be gained from smaller towns or villages - my idea of future LotR game takes place in a small place in eastern parts of the western Middle Earth during the rise of Sauron, where heroes need to defend the town against orc infestation.
Looking ahead: Conflict between two game styles
The next game I'm planning to start is Beasts & Barbarians, which by default uses very different style. Characters wander from location to another meeting very different situations. What makes this different from default fantasy is the idea that characters don't want to be boring and use their money in new equipment; adventuring is their job, and between adventures they spend most of the treasures they have found drinking, wenching and whatever they enjoy.
Now, I want to keep on using the BoO -concept that I've always found better than wandering around when running B&B. I still want to use B&B's own mentality in the game, and while I have some ideas about how to use them both in the game, I'd like to hear comments and suggestions how to do it. I'll post later what's the best solution I find for this, but until then, feel free to throw your suggestions at me!