Lie or lay

For some reason this line from the song has always annoyed me:

Like a bridge over troubled water, I will lay me down.

It never occurred to me it would be written that way, so I always assumed it was "I will lay thee down". Which is a little odd itself, but....

I understand the basic difference between lie and lay - you lie down, you lay some other thing down. Lay requires an object. It doesn't seem you can lay yourself down - that's kind of like lifting yourself up by your bootstraps. I guess the song doesn't quite have the right rhythm if you say "like a bridge over troubled water, I will lie down." It sounds awkward to sing "I will lie-ie down", or "I will lie right down", and songs don't always have the best grammar if they sound better that way.

Then I remembered the old prayer "Now I lay me down to sleep....".

What's going on here?