With a handful of good design mockups, then developers have something to aim for instead of a nebulous idea of what good design is.
We need designers that are willing to engage in the project and work hand in hand with the programmers.
In close feedback loops.
What you are suggesting will very likely lead to what
https://news.ycombinator.com/item?id=12218307
decided is bad design.
(Well, I don't really agree with him, but his opinion seems to be popular here).
What we need are lot of small incremental improvements based on a solid (evolving) concept, not a full rewrite each time a new UI mode shows up.
I suspect that would be extremely counter-productive. If you try to run some sort of design contest, and pitch it as being good publicity for the designers but not actually paying professional rates, then most serious designers will consider you a joke. They've heard every variation of this "spec work" argument before, they see right through it, and they will just consider you a time-waster. You should also expect to become a punchline on a lot of online design forums for a while, losing any credibility you might have had with other designers who didn't see the original contest.
The way I see it, if an OSS project wants to improve its presentation, it has the same two basic options as the programming side: either inspire people who genuinely value the project and are willing to volunteer their time and skills because they want to see it succeed, or raise funds somehow and pay professionals at professional rates.