Ah yeah, I recall that video, Mr Carlson's indeed.
You may want to be careful to avoid induced field effects -- you could very easily make an accidental Kelvin dropper, I suspect. Maybe kind of unusual for an antenna, being that the structure is usually wires not cages, and usually off to each side, not crossed; but weird shapes are used from time to time, and maybe with the right wind it could happen, hey who knows.
Oh, also that assumes some means of reference: it works when water drops off one electrode, in the field of the other; free raindrops/snowflakes are charged at whatever they start with from the clouds and can only be deflected by electrodes they don't touch. But that can easily happen with some colliding with the antenna then dripping on down the way.
So with that consideration covered, what's left should simply be charge deposited by particles in contact, which comes from the atmospheric potential. Which is, well, there you are. Hook up a meter and log away!
There's still one thing: charge removed by particles colliding or dripping off the element. If its voltage is held near earth, you can at least null the charge loss (or gain, when regenerative as above) due to dripping. Not sure what can be done about triboelectric charging though; especially if it's something like freezing rain or sleet and the elements get caked with snow/ice, and so you have snowflakes rubbing against an insulating layer, as well as sometimes landing in place depositing charge.
As for avoiding it, easy enough to galvanically ground it of course, which often happens incidentally with the use of a balun, tuner, or what have you.
Ed: heh, would be interesting to plot this as just some random ancillary data on one's weather station -- if so equipped. I'm not sure that it would really be very meaningful or interesting by itself, but it might be one of those oddball things that interesting correlations can be found on. Like varying with intensity of storms, or ionospheric activity or something.
Tim