Maybe skip it? I don't mean to sound like a Scrooge, it's just that spending any amount of money on a gift that deliberately is something the recipient doesn't want sounds rather anti-Mustachian. (This is assuming we're talking about the same definition of "White Elephant".)
I think "white elephant" has different meanings in different family/friend groups.
In my family, we've decided on "something you already own and no longer want/need, but someone else probably does." So it's good, useful stuff you just don't want anymore. Sometimes it's never been used.
On my husband's side of the family, however, they give crap for white elephant gifts. Like a half-used roll of toilet paper. Or a half-eaten box or cereal that expired a year ago. True garbage.
Part of the fun of the white elephant gift exchange is the "stealing" of the desirable gifts. So to me, the gifts have to be desirable for the game to be fun.