When Mario approaches the arena in Glitzville for the battle with Rawk Hawk, one of the crowd members calls him Jumpman. This is the name Mario went by in the original Donkey Kong (1981).
While in the X-Naut Base near the game's end, you propel yourselves upward into an air conditioning vent. Normally, you go to the left, but you can go all the way to the right until you hit the wall and another vent. Turn paper thin and fall into the chamber below. You'll be in a changing room. Open up the curtain (NOTE: Try looking at your Allies menu during this, too.)
A small toad in Petalburg will rave about his favorite video games if you talk to him. When you go into the town for the first time, he will talk about Fire Emblem for the Game Boy Advance. Later, around the time you make it into Hooktail's castle but before you beat Hooktail, he'll be talking about Paper Mario (2000) for the Nintendo 64. Finally, when you come back to the town after beating Hooktail, he'll tell you that he's playing Paper Mario: The Thousand-Year Door. All three of these games were developed by the company Intelligent Systems.
In the scene where Peach takes a data disk from Grodus' room and puts it in his computer, the image shown on the computer screen is the start-up theme for the Famicom Disk System, an add-on for the Famicom (a.k.a. Nintendo Entertainment System) released only in Japan. The Disk System used floppy-like disks instead of cartridges and helped to pioneer the concept of saving game progress. Some of Nintendo's most famous games, such as The Legend of Zelda (1986) and Metroid (1986), were first released on this add-on system.
Winning in-game the Happy Lucky Lottery isn't actually based on luck at all but on the date from when you bought the ticket based off of the GameCube's internal clock. The prizes are as follow: 4th prize is anywhere between 4 to 10 days, 3rd prize is 25 to 35 days, 2nd prize is 85 to 115 days, and 1st prize is 335 to 395 days. If you change the system's clock to a day prior and went back to talk to Lucky who runs the lottery, he'll know that the internal clock has been messed with. You can either admit to it or lie to not having tampered with it, however you'll need to pay 500 coins in order to reset the lottery, regardless. Daylight savings is not put into account. Buying a new ticket won't actually increase your chances of winning either but will instead reset the days back to 0 from when you bought it.