How about the probabilistic method?
There are plenty of elementary, self-contained examples to choose from, and it has a pithy slogan that's memorable enough even for non-combinatorialists. (Can't construct something explicitly? Then construct it randomly!) Best of all, it has a nice wow factor: While many undergraduates may be familiar with nonconstructive phenomena in mathematics, the fact that we need to resort to such to say things about finite graphs is rather surprising.