It does seem pretty evident that in this particular song the Mary originally referred to is Mary aka Miriam, sister of Moses and Aaron, rejoicing at the downfall of Pharoah's army - see my post above.

Of course it's quite possible that people singing it might have got their Maries amalgamated or transposed. That's the kind of thing that happens in folksongs.

Plus, of course, the exhortation to "Mary" not to weep and mourn is in real terms directed at the congregation singing the song.