There are a few reasons I think Eddie will decide to stay:
1) His upcoming contract is not awful. It's not like he would be accepting a deal Pruit worthy.
2) Although his shooting is unmatched in the league with his release and accuracy, not every team values a one dimensional guy the was the Celtics do. He opens up the floor for better scorers inside, whereas some teams like their big names (Lebron, Rashard, Vince, Carmelo, Billups) to handle the ball on the perimeter and work from there.
3) Someone else made the good point that Eddie has a family (we all see his kid at every game) and has been a journey man. Settling down seems important to him and he has that chance here.
4) Accepting his player options now gives him a better chance that the Celtics will either negotiate a reasonable extension, or possibly offer a decent deal next offseason to make him a mainstay.
5) The fans love him. The crowd goes wild with every shot he hits and he really feeds off of that. His intensity is right in line with Garnett, and the Boston fans love that and he knows it.
I think Eddie stays.