From Zelda Legends Wiki
Fire is usually connotated with the golden goddess Din, which may be why this spell is named after her. The spell may also use her power. Consider the creation account from the OoT manual: "Din, with her strong flaming arms, cultivated the land to create the earth." The ALttP manual says that the "God of Power dyed the mountains red with fire and created land." So either this spell's name only figuratively refers to Din's fire from the creation account, or the spell really does use Din's power to create flame.
"Din's Fire (6 MP) - Fire wraps around Link's body, protecting him from harm. This is very helpful when surrounded by a large number of enemies." (p.25)
Item description: "You got Din's Fire! Its fireball engulfs everything! It's attack magic you can use with (C)."
Great Fairy's description: "You can use Din's Fire not only to attack but also to burn things!"
In Toki no Ocarina, this spell is given the name "DIN no Honoo," or Din's Flame. "Honoo" means flame or blaze, but this kanji is usually translated to "fire" in OoT. The same kanji is used to describe Din's "flaming arms" in the Deku Tree's account of Hyrule's creation, so this may indicate a relationship.
The German version of OoT calls this spell "Dins Feuerinferno," which means "Din's Fire-inferno." In the French version of OoT, the item description says that you can call upon a "puissante aura de feu destructrice," or "a powerful aura of destructive flame."