Hello Bitcoiners,
Before I describe the idea, note that is is submitted to Finland's SoMePitching entrepreneurship contest (yes, an idea for a competition, submitted to a competition). Check out the entry and consider voting (link in Youtube description)! This was a rushed, crude job, but perhaps captures the concept a bit I hope. Hoping to land some seed sponsors and supporters and people interested in kicking this off IRL. 
https://www.youtube.com/watch?v=t2V2fnMx4CI&OK, as for the idea. Think of the TV show Shark Tank, but allowing anyone to participate, a social media component, and focused on Bitcoin and similar P2P stuff; for real cash and prizes and publicity. You may be familiar with similar competitions like the XPrize, MIT100K, etc.
Apologies if the video offends any geeks (I can because I am one) but I thought Bitcoin could use some off-forum/off-IRC promotion/investment and good stories. I think this is somewhat aligned with the reasons others had for starting a foundation. My experience in other industries with entrepreneurship competitions has been amazing and very valuable. I know there are some other small-scale competitions and stuff out there, like the Bitcoin hackathon. But I was thinking to do something with more exposure to the wider entrepreneurial & investment world and grow the Bitcoin economy. Hence the BTC Prize. Note the focus here is on decentralized, P2P money systems, not just Bitcoin, though Bitcoin will probably be a big part of this and the name is obviously a nudge/wink to Bitcoin anyway. 
This kind of competition would:
- promote & vet business & technical ideas
 - allow existing Bitcoin businesses to develop & connect with valuable resources, investors
 - put a positive spin to Bitcoin, in contrast to all the hacking, theives, scams, etc.
 - provide networking that builds a powerful incubation for startups
 - spur interest among students, young people, entrepreneurs
 - get Bitcoin in front of business professionals in law, banking, IT security, etc.
 - entice investment in Bitcoin infrastructure
 - media attention & advertisement opportunities
 - something for the Bitcoin community to rally behind
 - foster international communication & relations
 - opportunities for education and training
 - provide relevant experience for people seeking work within existing Bitcoin companies
 - volunteers and organizers will network with top names in Bitcoin, economics, etc. & travel the world!
 - spread ideas to the general public for BTC business ideas and tech
 - get people in the Bitcoin community out in the open instead of behind online personas
 - spur the liberty community to real agorism!
 
Obviously I think the organizers of such an event will have a great opportunity to connect with luminaries in the field and build on ideas shared by others. This could start small and then expand.
Some similar competitions and their "Social ROI":
X-Prize: A $10 million initial investment led to $100 million in spending by teams, which in turn led to a $1.7 billion investment by private industry. Now the field of private space exploration of space is about to grow exponentially, as a direct result of that initial $10 million.
MIT100k - facilitated the birth of 160+ companies w aggregate exit values of $2.5 billion, a market cap of $15 billion+.... generated 4,600+ jobs and received $1.3 billion dollars in Venture Capital funding..... multiple IPOs, acquisitions and success stories: Akamai (AKAM), net.Genesis(NTG), and C-Bridge Internet Solutions (CBIS), etc.
DARPA_Grand_Challenge - start of Google self-driving car
...
Of course I agree that there is still a place for underground, decentralized investment. And perhaps this competition can indirectly help make that more robust too by incubating better decentralized systems for the use of everyone. I could see this morphing into a more broadly focused agorist business competition down the line.
Incidentally the 50% purse/50% admin figures for fund-raising overhead were taken from Xprize itself. Perhaps this kind of organization could be leaner.