Instructions
Change your partner to Lady Bow, the Boo. This can be done in the menu before the fight with the Big Lantern Ghost.
Use "Smack," Lady Bow's basic attack, on the lantern. It will light up after she hits it.
Attack the Big Lantern Ghost with Mario and Lady Bow while the lantern is lit.
The Big Lantern Ghost will extinguish the lamp after you have attacked him a few times.
Use Lady Bow's "Smack" attack on the lantern every time it goes out. Mario and Lady Bow can only damage the Big Lantern Ghost while the lantern is lit. Repeat this process until the Big Lantern Ghost is defeated.