Here's an interesting article I found while poking around the Web for something else: Why W?, by Brian Hayes, from the March-April 2005 issue of American Scientist. W is not the president here, but the inverse function of x → xex. Apparently the name didn't become popular until the Maple software was written, because Maple needed a name for the function.
Empirically speaking, when I type random things I'm curious about into Maple, the W function is one of the functions that most frequently appears in its analytic solutions; Bessel functions are also quite common, which shouldn't be surprising, as are various instances of the hypergeometric series. Since so many familiar functions can be written in terms of the hypergeometric series, this is hardly surprising.
The coefficient of xn in the Taylor series for -W(-x) is nn-1/n!; the number of labelled trees on n vertices is nn-2 (Cayley), so it's not that surprising that W is involved in the enumeration of trees. In fact, W(x) is the generating function for labeled rooted trees. (n rooted trees correspond to each unrooted one, since we can choose the root from among all the verticecs.
Unfortunately most of these trees are very oddly shaped and would not look nice as a Christmas tree.