Auction sites like eBay work pretty well if you're looking for a hard-to-track-down piece of flatware. But can they work for lawyers' services?

Many attorneys have doubts, but a 21-year-old law student named Robert Grant Niznik is putting his money—or more candidly, he says, his parents' money—behind a website called Shpoonkle that will let lawyers bid for clients.

The...