In the ancient world, a name was not merely a label but an expression of character and an exercise of authority. 'In the name of' (en to onomati) means with the authority of, representing, or on behalf of that person. Prayer in Jesus's name, baptism in the name, healing in the name — all invoke his presence, authority, and character.
The NT 'name theology' is enormous. Jesus came in his Father's name (John 5:43). He made the Father's name known (John 17:6). Believers are to pray 'in his name' (John 14:13-14) — not a magic formula but praying in line with his character and will. Philippians 2:9-11 declares that God gave Jesus 'the name above every name' — the divine name that commands universal submission. The Great Commission baptizes in 'the name' (singular, not names) of Father, Son, and Spirit.