Items

Cactus Thorn: Successfully complete mission 10-1-3 (adds 1000 needles to DMW).
Tonberry's Knife: Successfully complete mission 10-2-3 (adds Murderous thrust to DMW).
Cait Sith's Megaphone: Find in a treasure chest in mission 8-4-3 (adds Courage Boost! to DMW).
Moogle's Amulet: Find in a treasure chest during mission 8-4-4 (adds Moogle Power to DMW, and levels all equipped Materia to Master level).
Magic Pot Tricks: Follow the orders of a Magic Pot (adds Item Mugger to DMW).

Materia

Ifrit Materia: Successfully complete mission 8-1-1 (adds Hellfire to DMW).
Bahamut Materia: Successfully complete mission 8-1-4 (adds Mega-flare to DMW).
Bahamut Fury Materia: Successfully complete mission 8-5-6 (adds Exaflare to DMW).
Odin Materia: Successfully complete mission 8-1-6 (adds Zantetsuken to DMW).
Phoenix Materia: Obtained during "7 Wonders Of Nibelheim" quest (adds Rebirth Flame to DMW).