I agree about it not being related to Lee and Leigh. Unless it is spelled Lea and used as a variant (pronounced exactly the same). So Lea could be a variant of Lee, Leigh or of Leah. Leah however doesn't have anything to do with Lee.
I think "weary" is a very likely meaning. In the bible Leah has "weary" eyes. However people fight whether the "weary" is to be translated as meaning "bad eyesight" or "ugly". We all know that the biblical Leah wasn't described as an attractive person. Her sister Rachel was the beautiful one.