Prof. Oak's Japanese name is Dr. Yukinari Okido. Ōkido is a Japanese family name and placename. Literally, big-tree family (Source) .
Thus, Okido is a common family name in Japan. In translation, however, this was changed to Oak, as well, a pun you could say. Oak sounds similar to Okido, and is a big tree. His first name, as said in the above mentioned source, is Samuel, which could be based on the Oak tree at Salem, New Jersey. See the similarity?
After that, it probably became a running gag, a sort of inside joke among developers. Note that it's not easy for people to remember unconnected names. Having them all related so had two purposes, namely, reducing amount of effort required to come up with the names, and two, to make it easier for players to remember. Finally, it gave a sense of familiarity to older players who noticed by themselves that they were all plant names. In a manner of speaking, of course.
That's my take on this. It could be different from the official reason, but this is how it made sense to me. All clarifications/ suggestions are accepted!
