A few factors contributing to his low BABIP...
1) This season he has a line drive rate of under 14%. A rough formula is that BABIP should be LD% + 12%, so that adds up to an expected BABIP of under .260. He's at .232 now, so it's not like he's been THAT unlucky. Even with his expected BABIP, he'd be a .220 hitter.
2) Weeks has a high rate of IF/FB. In other words, he has a loopy uppercut swing which produces a fair share of pop-ups--15% of his flyballs are pop-ups every single year, which isn't very good (in contrast, check out Michael Young and Derek Jeter, who consistently maintain a rate of around 5-6%). People who hit more pop-ups tend to have lower BABIPs, since they are putting the ball in play but in a way that is virtually guaranteed to be an out.
Overall, I am very skeptical that Weeks will ever learn to consistently hit for average. Like an earlier poster said, one of these years he might get lucky and hit .280. But that's a pretty terrible reason to draft him--"Oh, if this is the year where he enjoys a mathematically improbable run of luck, then this will be a great pick!"