I can't seem to find any etymological root for the hyda- in hydathode. I expected the water-relater structure to be called a hydrothode, but it just isn't!
The etymology of hydathode, according to the Collins dictionary, is "way (hodos) of water (hydat-)". So the correct way of dividing the word is not hyda-thode, but hydat-hode.
Also, "hydat-" is the genitive of the Greek word for water, "hydor" (literally "of water"), whereas "hodos" is in its nominative form ("the way"). If you want to say "water way" in ancient Greek, you have to use nominative+genitive, not nominative+nominative, but there might be also euphonic reasons to why is not *hydrohode.