One approach I take is to think about a few different things:
1) What is the chord progression of the song?
2) What orchestration will sound good with it?
3) How big/what should my part be?
When I first start trying to write something, I might just play each chord on every beat, or once per measure to get a feel for it. Then I think about which inversions sound good, and how one leads to the next. I also think about how to vary it rhythmically.
Usually I have an idea of what sort of sounds I want to use. In your case, you're already thinking of organ, so I'd start with that. I might also try something that's completely the opposite, just to see how it sounds and give me new ideas. Sometimes I'll record something into the sequencer and just switch patches as it plays back to see what it sounds like, but usually I have some direction I try to take it.
And of course, you don't want to upstage the main performer. Is the piece written to showcase a singer? Guitar? Something else? I usually try to stay out of the range of the lead so as to not overwhelm it.