Alright, usual rules:
-One game per IP
-Have to have played the game
-Going to equate "start screen" with "menu screen" in some cases, but this doesn't include intros (which I'm pretty sure there's already a thread on)
-Advance Wars: Black Hole Rising ("move out!")
-Army Men: Sarge's Heroes (music)
-Banjo-Kazooie (Banjo's house)
-Batman: The Telltale Series (atmosphere/ambience)
-Battlefield 1942 (pumped up)
-Command & Conquer 3: Tiberium Wars (atmosphere)
-Conker's Bad Fur Day (humour)
-Diablo IV (atmosphere)
-Ecco the Dolphin: The Tides of Time (music/ambience)
-Fire Emblem (really any game I've played, the theme alone sells it)
-GoldenEye (turns out files and dossiers can be fun)
-Golden Sun: The Broken Seal (music, melencholia)
-Halo 3 (1-2 are close seconds, since they're all variations on the same thing, Halo 1 has grandiosity, Halo 3 has tragedy, Halo 2 is the best score by itself)
-Heroes of the Storm (pumped up)
-The Lord of the Rings: Return of the King (stepping through history in a sense)
-Marathon 2: Durandal (solely because of that score)
-Metal Gear Solid (sheer atmosphere from the theme/graphics)
-Mission: Impossible (yes, the N64 game, and yes, the start screen got me to accept the mission)
-Perfect Dark Zero (theme/ambience)
-Pokemon Gen. 1 (theme/nostalgia)
-Section 8 (eh, sort of? Game is balls, start screen's okay)
-Sonic Adventure 2 (really, there's too many competitors, going with this one for OST)
-StarCraft (ambience)
-Star Wars: Rogue Squadron (if you equate start screen with mission and fighter select, then sure)
-Super Mario 64 (why yes, you CAN pull Mario's face around)
-Warcraft III (ambience)
-Warhammer 40,000: Dawn of War (OST)
-Xenoblade Chronicles (music, ambience)
I'm also going to give an honourable mention to Terminator: Salvation, in that while I haven't played the game, its riff on the Terminator theme is excellent.