As the others have said, lye does not "become safer" with time unless there is something for it to react with. In ~small~ quantities such as tiny drops of lye or a thin film in a container, lye can react with things like CO2 in the air, given enough time. So in some situations, yes, lye can "become safer", but again only in the right circumstances and given enough time.
That said, if you DON'T KNOW, then DON'T ASSUME. Treat any lye residue, however large or small, as if it is just as reactive as when the lye was freshly made. The advise that has been given to use reasonable safety gear and rinse well with water is spot on correct.
ETA:
"...Got a thick trace and then (oh no!) poured it back into my lye water measuring cup for easy pouring. *Face palm* After pouring into my molds, I realized what I'd done. I'm going to guess this soap isn't safe to use seeing as there could be unsaponified (is that a word?) lye in it? ..."
Well, the soap batter you were messing with had a lot of unsaponified lye in it too. From what I gather, you poured soap batter that had a lot of strong lye in it into a measuring cup that contained a small residue of the lye solution. If that is the case, I don't get your concern.
Maybe there are some important facts missing -- for example, did you have a lot of lye solution in that measuring cup???
ETA2:
And you're going to get a lecture about mixing lye in a glass container. Use stainless steel or lye-safe plastic only!