Here's a little cheat for you. Nine times out of ten, if you have a professor or teacher in a film, that person will be a mentor character. There are exceptions to the rule—looking your way, Peter Venkman—but Professor Yamane is not this exception. He's a mentor character all the way.
Yamane's hired by the Diet Committee to investigate what's happening in the oceans around Odo Island, and his initial findings lead to the discovery of Godzilla and its origins. Through him, the characters—and by proxy the audience—learn that Godzilla is a marine reptile that's been around since the Cretaceous period. We also learn through him why the creature has developed such an attitude recently:
YAMANE: It was probably hidden away in a deep-sea cave, providing for its own survival and perhaps for others like it. However, repeated underwater H-bomb tests have completely destroyed its natural habitat. To put it simply, hydrogen-bomb testing has driven it from its sanctuary.
After providing this initial study, Yamane is kept on by the government and Defense Forces to advise what to do about Godzilla.
Can We Keep It?
In many monster movies, the feelings of the characters toward the monster are pretty straightforward: kill the beast or it'll kill us. Come on, it's not like you can reason with a building-sized dinosaur trouncing its way through your neighborhood. It probably can't even hear you from way down there.
But Yamane adds some depth to our understanding of Godzilla by being sympathetic toward its existence. As a zoologist, he sees Godzilla as a valuable specimen, something that can teach us about our world. As he tells the president, Godzilla can't be killed, so their "priority should be to study its incredible powers of survival."
Unlike a Frankenstein-like scientist, who disregards humanity for his research, Yamane has humanity's interests at heart. He sees the creature's ability to with stand radiation a once-in-a-lifetime opportunity to help humanity itself survive the nuclear age. It's also evident that Godzilla itself is a victim of the nuclear bomb, so our sympathies go out to both the creature and Yamane's unpopular view.
But the fact that no one else agrees with him leads him to depression and bouts of anger, like when he yells at Ogata for suggesting Godzilla's too dangerous to be allowed to live. While there's truth to what Yamane believes, we've got to give the nod to Ogata on this one. There just isn't a zoo or animal sanctuary large enough to safely study Godzilla.
You saw what happened to Jurassic Park.
Too bad though. The viral videos would have been amazing.