It's been started by @RWAKissofDeath , and it's more of a networking thing than something that's going to get you an agent. There's no "benefit" to doing it, other than getting followed by hordes of writers. It's AMAZING for attracting wannabe writers. Put up a #1LineWed tweet and watch those pretty numbers fly!
The good part of it is that you get lots of amateur writers re-tweeting your tweets and following you.
The bad part of it is that they are only re-tweeting your tweets and following you because they want to be re-tweeted and followed by you.
But that's not the best part! The best part is, for the rest of the week , you can look forward to endless notifications from Twitter telling you that "@CrazyCatLady369 has favorited a tweet you were mentioned in", as people send out twitter thank-yous crammed full of user names, and re-tweet all the messages with their names in them and favourite every instance of either or both for as long as physically possible. What fun! We all love notifications, don't we?
The nice thing about it though, is it's a great way to build up your "following" numbers, as long as you don't mind an audience that never pays any attention to anything you tweet.
Unless it has #1LineWed in it - in which case they'll re-tweet it with their dying breath.
>Removes tongue from cheek.
EDIT - Oh, every week has a theme. Last week it was coffee. The week before it was "Something from page 2 of your novel". Most people try to stick to the themes, but quite a few don't bother. They all get re-tweeted anyway, because nobody ever reads what they re-tweet as long as it's got the right hashtag. I proved this last week by tweeting "Fart in a bottle #1linewed" and THAT got re-tweeted.