From Welsh

The name Cadi is best understood as a variant or modern spelling connected to Caden. The root name Caden has origins in modern Celtic-influenced name and is generally associated with the meaning battle or companion, though origin is debated. This makes Cadi a creative or alternate form that carries the sound and general symbolism of Caden, while giving the name a more distinctive spelling. Because Cadi itself may not have a separate ancient etymology, its meaning is most safely traced through the better-documented root name Caden.