Twilight Paths story writer
I'm another player writing up his DM's campaign. We've found a few things that helped lots.
1. Getting "flavor text" direct from the DM. If you're a player and you don't remember exactly waht a room looked like, what an NPC's name was, etc. just ask the DM to copy and paste that stuff and send it to you. Other great stuff from the DM is stuff like back history on the world that your characters discover in their adventures that you can copy and paste.
2. Having some of the conversations online in between games. This has come up a number of times, when the group was deciding where to go next. This way there's already a written transcript that you can usually take, edit a bit to get it all into the same writing style, and use directly.
3. Whoever's writing up the story hour should definitely take notes, though they don't have to be too detailed, depending on your memory. I usually jot down the names of people we meet, just so I don't have to pester the DM as much later, and so that I remember for later in the session! I'll also write down some basic descriptions of where we went and what we did, in case I wind up not writing stuff up until a couple of weeks later. Any good quotes from any character I try to jot down, as well as anything I think my character (the author of the story hour) would notice.
One other thing that's worked well for us is to have me write up the story hour, and not the DM. Most of the story hours here are written by the DMs, but we found that the DM already had his hands full running the campaign, and that when the DM tried to keep a journal he had a hard time editing stuff the right amount to not give anything away. Piratecat and the rest of the DM storytellers do great jobs though. I suppose if you're the DM and are also writing the story hour, then as long as you wait a few sessions before you post, you probably won't give anything away.
One final thought... I'd start by writing up every session for several sessions before posting. This does two things. First, it gets you comfortable with your writing style and voice, giving a more consistent feel to your story. Second, it lets you post a ton of stuff up front, to get interest going, before getting into the two week waits in between writeups while you wait for the next game session.
... guess that's all I got.