My point is that is isn't just a function, but a measurable one (with weak restriction on the measure space it has as domain) so we can't call it just "function" as you were suggesting.

The actual name might be debatable, but a shortened name rather than the full definition definitely makes sense here. Random variable seems like a good choice of name to me, thinking about the intuition we're formalizing with it.

But you propose to simply relabel as function, which doesn't work in general because random variable corresponds to a specific type of function. You could compromise by calling it a probability function, but then you start to collide with other uses of that word.

I agree random variable is awkward, though. I always avoided stats courses because it's full of so much jargon that collides with nomenclature used by mathematicians.