I don't know what gave me the idea.. lol, I remember watching a video about this a while back.. but just remembered it today out of the blue.
Basically, it'd need to be small scaled for now, but maybe ideas for the future companies! (and/or countries!) But I can't believe it's seriously gotten to this point... a garbage patch bigger than the size of Texas, in the Pacific Ocean! They find fish and birds as well, their stomaches full of the plastic they ate. (I mean, FULL.. it's more plastic than meat inside of the fish!) But anyways..onto the "Idea":
An aquatic robot, whether autonomous or remote controlled, that has the ability to scoop garbage and sift/filter/not sure and vacuum the garbage to a container, or sorting dock to be checked for fish to be thrown back etc.
Of course, we're trying to help SAVE the environment (at least part of it), and not kill all the fishies as we're cleaning up!:)
Here's an example of an Albatross.. a HUGE bird, the plastic they mistake for food and eat.
And here's just an example.. I'm not sure if this in the gyre itself.. but was compared to how bad it is. (They said the Gyre is much worse than this)
Ideas for garbage collection:
Initial thought.. A net. Of course, fine enough to filter alot of the broken plastic (which is as small as some plankton) but of course.. we can't get it all, the more the merrier!
Second thought.. A vacuum. And filter the water that's coming through.. much like those huge ships they use to move sand in the oceans.. suck the sand up and fill it's hull with sand. A video below shows the basic idea.. but for garbage! (second video)
Third thought.. A dredge. (or huge shovel) but this seems like the most.. time consuming, but possibly the safest (for the fishies!) (Third video)
Fourth thought.. Giant bucket. (or huge blanket type thing, maybe even a filter) For example, like they use on Helicopters when they're fighting fires.. have the huge loads of water on the bottom. But instead of picking up all the water, have it finely filtered, and drop it in the "sorting area" to try sort out the wildlife. (here's a link of what I'm talkin about, http://en.wikipedia.org/wiki/Helicopter_bucket)
Fifth thought.. [[YOUR IDEA HERE]]. This is where WE (the WORLD) needs you.. I'm sure all these have already been thought up.. and looked at as possible ideas.. But we're a smart bunch of... smart people! I think we could come up with some awful good ideas!
I'm not really sure how to make it a "competition" but maybe a "Group Project".. or you could set up a little pool in your backyard or bathroom, haha.. get some (FAKE) fish and try out some ideas. (DO NOT use Goldfish either, they're a mess! :D Not sure how to emulate the garbage.. maybe just some small pieces of paper, or broken up styrofoam cup. But hey, ideas are good too!
Of course, it may not be the "Problem Solver", it will take ALOT more than just some robots/ships/whatever you want to call them.. but one thing at a time!
(Robots that can crush and sort plastics maybe to help recycle better? Anything works!)
https://www.youtube.com/watch?v=uLrVCI4N67M
They’ve banned plastic shopping bags from being given out at supermarkets and stores to hold your groceries and purchased items, you can still buy rabbish bags 
hehe but its still a huge step forward as so many plastic bags were given out at stores and never reused.