Here are some deadlines:
The Oregon Space Grant Consortium Education Resources Blog seems to be keeping track of a lot of student space prize deadlines:
PISCES 2009 Student Design Competition - Notice of Interest Due By January 31, 2009
2009 RASC-AL Competition for College Students - Deadline February 6, 2009
Great Moonbuggy Race Competition - Deadline February 1, 2009
I'll add to that list:
8C Business Plan Competition 2009 - Step 3: Submit Business Plan Executive Summary (MANDATORY; by 2/6/2009) LAST DATE TO ENTER! (I'd check on steps 1 and 2 as well). This one used to be called Lunar Ventures (a name which I find a lot easier to remember).
UKAYRoC - Registration is closed; Qualification Closes 2 March 2009 and UKAYRoC 2008-2009 Final 2 April 2009
.. and here are some Facebook prize pages:
University Rover Challenge on Facebook
Team America Rocketry Challenge on Facebook (see TARC is on Facebook! from the TARC Blog Archive).