Well, you can win a little more than $5k.
http://www.avc.com/a_vc/2010/12/nyc-big-apps-version-20.html...
"One of the winners [of last year challenge] was a team of 6 who quit their jobs on Wall Street, are in dozens of cities, received millions in VC investment (at a multi-million dollar valuation), are profitable, and recently rang the closing bell on NASDAQ."
I think BigApps compares a little bit with the Google summer of code where you get pay to make something cool and if it's cool enough it turns into a job/business.
The YC model idea sounds interesting, though.
Last year had quality entries, and I'm pretty sure the grand prize was the same.
You can see last years entries here: http://nycbigapps.s3.amazonaws.com/index.html
Why not use the YC model? Accept applications, pick the 10 best ideas/teams, and give them each $10k and 3 months to build the app, in exchange for 5% equity. Have a panel vote after 3 months to award a $50k bonus (additional incentive to make a great app).