What does “in the name of…” actually mean?
Putting all religious contentions aside for the sake of our language, the etymology of name offers a good place to start understanding:
Old English nama, noma "name, reputation,"
from Proto-Germanic *namon
(cognates: Old Saxon namo, Old Frisian nama, Old High German namo,
German Name, Middle Dutch name, Dutch naam, Old Norse nafn, Gothic
from PIE *nomn- (cognates: Sanskrit nama; Avestan nama; Greek onoma,
onyma; Latin nomen; Old Church Slavonic ime, genitive imene; Russian
imya; Old Irish ainm; Old Welsh anu "name").
We've all experienced the power of namedropping in our lives. People respect us and our opinions if they believe we are connected to someone with greater reputation and authority.
In all cultures, people of authority have always lent their reputation and their authority to their delegates. The founders and leaders of religious movements use the same delegation strategies as the founders and leaders of nations. The English phrase in the name of simply asserts the reputation and authority of another person.
English Reports Annotated - Pages 1505-2672, 1505, page 2048:
...an action on a board given to trustees of an industrial society
before the act may, after registration under the act, be brought in
the name of the newly -incorporated body.
Victor Hugo's Dramas 1519, page 364:
Richard Varney, in the name of God and Saint George we dub thee
The Newe Testament in Englishe Translated After the Greke, 1553:
And he that receiveth a righteous man, in the name of a righteous man,
shall receive a righteous man's reward.
Calendar of State Papers, Foreign Series, of the reign of Elizabeth: preserved in the State Paper Department of Her Majesty's Public Record Office, Volume 7, 1564:
Smith and Throckmorton in the name of their Mistress demanded the
ratification of the Treaty of Cateau Cambresis
An Open Letter to the Christian Nobility of the German Nation, by Martin Luther, 1520, Translation by C. M. Jacobs, Page 94:
The complaint was made at Worms (1521) that it was impossible for a
German to secure a clear title to a benefice at Rome unless he applied
for it in the name of an Italian, to whom he was obliged to pay a
percentage of the income...
We introduce an interrogative with the emphatic: What in God's name, or its metonym: What in heaven's name. That emphasis poses an implication to the listener: I have a right to ask this question, and you owe me an answer!