Well, I'd suggest you start by looking at the list of
densities of common materials on Wikipedia. Then you can sort them in order of density by clicking on the little arrows at the top of the list.
Start at the top of the list and work your way down. The rules prohibit you from using dangerous or harmful materials, so Osmium is probably off the list. You also have to be able to find a common source of the material, "available to all teams". So unless you can find an easy way to purchase Iridium, it's probably off the list, too.
But technically, anyone can buy Platinum.
http://www.monex.com/prods/plat_maple.html
If you want a non-ridiculous answer, however, I'd look a bit farther down the chart.
You might also want to check out the densities for Brass and Bronze, common engineering alloys of copper, to see how they compare to the other densities listed on that chart.
Jason