Which of these three beloved games, each of which transitioned their series to the third dimension, do you think is the best, and why?

The best game, or the best adaptation?

In terms of best game I'd say Mario 64, but that's not a faithful translation to 3D. Mario 3D Land would be the first Mario game that really translated the 2d ones into 3d. Metroid Prime is probably the most faithful adaptation. Obviously, they can't directly carry over everything from 2D Metroid, but they did a surprisingly good job.

But Mega Man Legends is the best. Fight me.

I meant best game overall, but if adapting the series is an element of that for you, that's fair as well.

In a straight comparison, I can't really do that. Mario 64 had a small hub and was very level based within the paintings, OoT was about as open world as a game could get back then. And Metroid Prime 1 was a GameCube game, which has a massive power advantage over the 64.

The only way I could compare them is who made the best leap forward compared to their mostly powerful and most recent predecessor. So...

Super Mario World --> Mario 64
Super Metroid --> Metroid Prime 1
A Link to the Past --> Ocarina of Time

And... My head just exploded. I still can't lol. They all were such great sequels that evolved their series so much. None of them are far ahead of the others.

64. Mario 64 still stands as an amazing game. It set so many bars for 3D games. Oot did its best for the era to make a 3D Link to the Past work but I feel a lot of mechanics have aged poorly. Take away nostalgia and OoT of those 3 feels the most like a product of its time. Prime did a phenomenal job adapting Metroid to first person. Still a great game. Held back a little by stiff controls limited to a Gamecube controller. Great game but didn't change the landscape.

Mario 64 didn't just adopt Mario into 3D but changed the industry. OoT and Prime for the time were just good at adapting them into 3D IMO.

I like all these games so I am enjoying how this is a friendly battle where everybody wins.

I'm not going to say my opinion is in any way objective, or even pretend it is. It's all subjective. We all have our own tastes. From my POV, I don't find Nintendo got it right with any of their first attempts on those particular games. I'd have voted for Lylat Wars or Mario Kart 64 if they were on the list.

I didn't feel Mario transitioned over to 3D well until Super Mario Galaxy.
Legend of Zelda until Breath of the Wild.
Metroid, still waiting.

Of those three, Ocarina of Time came by far the closest to me feeling like a good transition. I still don't think I will ever go back to that one, as I don't enjoy wading through 3D dungeons for hours searching for keys and switches. I much prefer the more varied and fluid feeling experiences of the official 2D games and Breath of the Wild.
Mario 64 did impress me a lot with its water effects when I first saw them, and the freedom and such, but the novelty ran off quickly for me and Mario 64 was my least favourite of Mario games to that point - well, including the SMB trilogy and SMW as I'd probably take 64 over the GB platformers (maybe, it's been a while).
Metroid Prime simply does not feel like Metroid for me, not a big fan of FPSs either. Metroid was the other open world exploration game on NES beside Zelda, except with terror and platforming, until Metroid can get that right in 3D, I likely won't feel differently.

For me it's Mario.

I'm not really into Ocarina of Time or Metroid Prime.

Ocarina of Time for me. Haven't played Metroid Prime, but I guess it aged better than the other two.
Super Mario 64's camera controls are almost unbearable to me, so by default Ocarina of Time. That said, ignoring the camera, OoT is still the better game.

I forgot about the camera. Yeah, I think that really ruined the experience for me. Fiddling to get the right angle, and then when getting it correct it was still wrong :D

I did finish the game, but the camera was by far the most difficult part of the game for me. Is this fixed in any of the later versions? I might give it a try.