Yeah, that's the exact entry I found at thinkbabynames.com. I didn't really accept that, because Leith is not really a variant spelling of Lethe (LEE-thee), but an entirely different name.
However, the similar pronunciation might explain why the name would be found on both sexes, better than to say it's simply a surname used as a first name (since Lethe seems likely used more for females). Although I'm not sure about that, either.
Another connection (via sound) might be with Letha & Alethea (linked).
Can you tell me where (geographically) you met your Leiths, and roughly when they would have been born?