How about a good visualization of biological networks? http://www.hiveplot.net/ explains very nicely the problem of the 'hairball'; whether the proposed hiveplot is really a break-through remains to be seen, but there must be more not yet invented alternatives to the hairball, there is definitely room for improvement.

Some kind of better-than-a-venn diagram way to represent set intersections for multiple sets. And yeah, there's this thing, but...

Basically any biological problem where you think some cool visualizations could be helpful is open. The vast majority of visualizations for biological data is pretty rudimentary in my opinion. Complex networks like Micheal Dondrup suggested is a particularly striking example. I'm very interested in dashboards myself for exploring genomic variant information myself. Most of what I deal with is in databases and spreadsheets where we set filters and look through results but I've though many, many times about complex dashboards for visually filtering, sorting, and looking at the data in a variety of ways simultaneously. I'm very open to collaborating with new people on this sort of thing.

I tried coming up with a better Venn some years ago, called an "Eulergrid". I would love someone to make my Eulergrid plots d3-enabled, adding some event handlers to turn sets on and off, resort sets by various criteria, highlight subsets by some subgrouping, etc.