After reading a few of the other ideas that people have posted about the gods being like people with issues and downsides I think it woud be nice to play off of that. Having gods with aspects that relate to them that go together and then one that does not. Having that one thing that they are forced to be that throws off their balance as a god. Like god of war and destruction has to also be sometimes the god of compassion for the weak. Making them have to change gears so to speak so that they can't be all representative of a aspect with out some kind of flip side to it that they have to do now and again as it is part of their nature.
I would like to see a god that is newly born into the pantheon of gods. The kicker is that he/she is a completely new god brought into existence by the other gods to "complete" them in some way or balance the odds somehow in a god(s) like dispute. Being newly born the new god has no sides to take or a pre-determined role in the world. After his/her creation the other gods that have summoned him into being get into a heated argument about the new god's place in their hierarchy so that they begin to ignore him. Since no one is providing a cue to what to do next the new god decides to go and see the world for themselves. In doing so he runs into our adventurers and begins to follow them of and on as they do interesting things that draws his attention long enough for him to check in on them. Another god setup that I think would be interesting would involve a god of some kind of magic school or sect that has been removed from godhood and made mortal. They would retain most if not all of the godly intellect from their previous postion but none of the devine power to act on it. The twist is that they would be banished to the mortal plain to live out their now mortal life as they see fit but they are tied to the new god that has been created/raised to replace them and required to "train" them in on being the god of what ever they were before. Along the lines of a advisor to get the newly placed god up to speed so to speek. The now mortal previous god is at the beck and call of the new incarnation of their previous self but also is a representative on the mortal plane for that god and/or church. Along the lines of the High Priest for a god but not really willing but is the head of the religion for the god as being the previous god. It could turn into a situation where the followers are a bit uneasy and the faithful are a bit shaken by having the previous god around and speaking to the new god in less than reverent terms. Just some thoughts that could help drive some stories.