River water level

I would say not - the obvious way would be to have a micro-controller do the counting, and send the data back down the wires as a serial data stream - you could power it up the same cable as well. It would probably be wise to take multiple readings, and send the average back as water is rarely flat.

I seem to recall that existing water height systems measure the height in a tube, in order to reduce the surface movement of the water.

It's an EXTREMELY common operation, measuring water height (or depth) through a fixed size channel, in order to monitor the amount of water in the local river system - the data is usually sent back wirelessly, and I'd like to imagine that it's all powered by a small water turbine (but I've no actual knowledge of that).

Instead of measuring the height of water in a tube, measure the pressure in a "sealed" tube (at the top).

Interestingly, after posting yesterday, I did a little googling - and there's a site that lists all the UK river monitoring sites - and the ones I know anbout are listed on it. There's 48 in Derbyshire where I live, and you can check the river levels and flooding risks.

Not a problem where I live, as I'm fairly high up

Measuring the distance from above and taking an average with a micro is a good way to go.
Or maybe a non contact capacitive probe in the water.
I played around with the idea of a time delay reflectometer for measuring levels, just with a piece of uninsulated wire, but soon found the development time was way more than I was prepared to do.
If you want to transmit a digital signal 250 foot, and your not using a micro that can error detect then I'd suggest to use some 50ohm coax and impedance match it either end.