When I try to play Morrowind, I get the same huge annoyance: having to type in a new name for every non quick/auto save that happens. Ii just started plugging in random letters.

What I'm looking for is something that, in essence, mimics the "save feature of Oblivion, where it automatically applies a number in chronological order. Does such a mod exist?

Are you playing on the pc? I'm assuming you are, since you're talking about mods. In that case, you can just click your old saves to use that name. (Not sure how the XBox version works, but probably has something similar.)