First shown at Spaceworld 2000, Luigi's Mansion is the first game for the GameCube. As in the case of every game, it had many changes from its earliest form and the beta version is highly sought by fans, but like most Nintendo prototypes, it has yet to be leaked.
There is a lot of surprising information dealing with the development of this game, such as the original idea of it being in 3D. This feature was removed because the GameCube add-on would be too expensive. Luigi's Mansion also had removed boss ghost battles, including an Australian Hunter that would threaten to kill Luigi and mount his head on the wall (the fight would have been most likely a variation on the toy soldiers, but with only one ghost, and was meant to take place in the Safari Room) and a chef ghost that threw tomatoes. Other removed ghosts include the early version of the bowling ghost, that would leap out and scare Luigi, leaving him with only 50 total hp instead of 100. Since the early version used the in-console clock to time it, and had to be beat in 24 hours, this would have added a large amount of strategy in avoiding these ghosts. But it was only used in the E3 demo, so it would've most likely been a timer until you would stop playing. There was a ghost tracker, adapted to the boo tracker, that would have made these ghosts easier to catch. There is also data left in the game that suggests that you were meant to walk to the mansion at the start of the game, instead of having a cut scene. Dr. E. Gadd's lab is fully mapped with the training room, portrait room, and ghost portrification room accessible from the main room. A ladder led up through the roof and out onto the lawn; the whole area can be explored using a gameshark, but no longer leads outside.
Although there are rumors that this game started development on the N64, this is the first version ever shown off to the public and thus the first version of the game. There is only a demo clip of the game, and a very short clip of 3 ghosts playing cards. A shot of the ghosts sitting at the table, facing forwards (unlike in the clip) remains in the game.The reel of footage includes a ton of beta game footage and is really only to show off the graphical capabilities of the system.
SpaceWorld 2001The second time the game was shown off it was in playable form, and was very different from the final product. It featured a coin counter, a heart gage out of /100, and many other differences. At this point, the game had a doll house like design aspect, and it is said the mansion would vanish if the gamer ran out of time. Boos were more like the mice in the final version, only getting in the way, and the game looks darker.
Notice the regular ghosts practically look like the ones in the final; there was a version some point before this that used radically different designs. No footage is known of that version. Notice the purple scare ghost and what it does to the life counter. Some footage from this version (in other videos included) has a gauge for how much you can suck up in game. It seems there you could only hold a ghost so long before the poltergust would catch fire. The Game Boy Horror at one point acted like a camera giving a POV shot. This still sort of exists in the game, but you can't walk and use it at the same time, and is used to scan items. It would have taken a lot of power for the continuous render, and thus was cut. It can only been seen in screenshots from Nintendo Power.
The E3 versions look a lot like the SpaceWorld 2001 variations, and probably is.
Both discuss it pretty well, and helps explain a lot of stuff removed, and both are very different explanations of the betas.
Exploring E. Gadd's lab
There are a lot of claims of people saying they own the beta, and yet there is no more footage available, so it seems it has never been released. The Japanese version has some beta elements left in, as it wasn't as finished as the NA version, and the European variation has more elements added into the hidden mansion.
There is also a claim that there is an unused safari room ghost in said area. This comes from a Nintendo Power issue, but is false mainly because the issue it was mentioned in was released after the game released in Japan. There is also no trace of it in the files of the game, as well as no footage of it. So there is no way in Earth this could be true!
The Cutting Room Floor article, with a ton of unused graphics and such 
The Mario Wiki page on the subject