For example, I know Mario Party has a few songs like "Playing the Game" that is a rendition of the original mario bros 1 theme.

Here's the thing, I got the Mario Galaxy Soundtrack (awesome) and I was looking specifically for songs in other mario games that have the themes (or even HINTS of themes) that were created in the original mario games (1,2,3, even super mario world) so that I could compile them. So...can anyone pinpoint any Mario Game (mario party's, newer mario karts, etc) that had songs that you recognized from the oldschool? Specific song titles would be best. =)

I'm almost sure that Super Paper Mario has some classic Mario references in it's music, but I don't know which songs.

All 3 3d Mario platformers use the underground theme (It is used twice in Sunshine as the Delfino Airstrip and in the Sewers, and as Purple Haze Maze in 64) and Strikers Charged has a funky remix of the SMB invincibility theme

Super Mario Galaxy has a Remix of Bowser's Road from Mario 64

New Super Mario Bros. has the Wing Cap Mario music from Mario 64 for when you pick up StarMan.

Just remembered that the chapter opening in Paper Mario is a quick reference of I believe the overworld theme from SMB1. Not sure if this carries over to the sequels.

Super Paper Mario:
Lineland Road
An Unrivaled Battle
Nostalgic Underground
Bowser Battle

Partners in Time:
Tutorial (or, Super Mario Bros. Theme (remixed))
Peach's Castle

Paper Mario: The Thousand Year Door:
Super Koopa Bros. (3 different tracks, plus related fanfares)
Shadow Palace

Okay, I'm out of gas. I hope this gets you started!

I too would love to find all of the underground remixes/re-versions in all recent mario games. I know of 3 so far.

Mario Golf 64 (or toadstool tour, I forget) - Birdie Putt
Mario Party - Option House Theme
Mario Party - The Room Underground

Super Mario Galaxy actually has more than only the Mario 64 Bowser Stage-music.

I also recognized Underground theme, Airship theme from SMB3 (Airship Armada), one of the Overworld themes (I think it's called Overworld 2) from SMB3 (Sweet Cake Galaxy), and the original Overworld theme is in the Toy Time Galaxy.

New Super Mario Bros. also recycled some of the original themes. Like the Ghost Mansion from SMW, or the Underground theme from the original game.

Super Mario galaxy 2 has many Super Mario World themes like the Haunted Mansion or the regular level theme. It has Starman theme (wich they call rainbow mario), a Super Mario Bros. 3 Theme... mmmmmm...

I think there were others but I cant remember right now.