I was watching the nature documentary series “Planet Earth” last night where it was stated that over 300,000 gallons of water flow over Iguazu falls every second.
I thought to myself, “how do they measure that?”
So… How do they measure that? 
Take a cross section of a river (say, 1 foot wide) measure the volume of that cross section, let’s say it’s 1000 gallons. Find out how long it takes water to get from one edge of the cross section to the other, say .5 seconds.
So that’s 1000 gallons every .5 seconds times 24 hours = 172,8000 gallons per day. Also, you’d probably want to measure the time from one end to the other kind of in the middle, I’d assume the top flows faster then the bottom.
That would be my idea anyways.
More likely a series of flow speed measurements at various points across and deep, someplace where it’s convenient to do that. They could then create a model of the flow speed and integrate over the area.
If there is someplace where the river has a long run of a stable simple cross section, or if it passes through certain shapes of weir (unlikely for a big river), there are published rules for estimating flow based on depth and slope.
Simply erecting a hydroelectric dam and measuring the electric output would work, wouldn’t it? Except for the dam itself, all you need is a voltmeter, an ammeter, and a resistor, right?
To supplement Napier’s response, use a Flowmeter.
To add a bit to what Napier said… Since the flows vary seasonally and with rainfall, the flow is measured at various stages, or water surface elevations, and a stage discharge rating curve is constructed. A stream gauge is installed or a stable reference feature is painted with a numeric scale to indicate the water level. Then flow can be estimated just by knowing the stage. The stage discharge rating curve has to be verified every so often. The USGS maintains a streamflow website with flow info for US rivers.
What about a Flowbee?
It could. But to determine the appropriate turbines and generators to install, you’re gonna need a decent estimate of the flow.
Don’t forget turbines, generators, wires, transformers, etc.