Eugene Nida (1947, p 178ff.) explains this way:

“The biblical attitudes toward human personality are of great theological importance. There is, however, only one word which produces any considerable difficulty in other languages. This is the word ‘name.’ The great difference attached to the significance of the name of a person in the Bible times in contrast with our own culture is very important. Note such phrases as ‘whatsoever ye shall ask in my name,’ John 14:13, ‘believed in the name of the only begotten Son of God,’ John 3:18, and ‘life through his name,’ John 20:31. These expressions are generally difficult for us to understand, for the word ‘name’ does not mean the same to us as it meant to those of Bible times. To them the name was the symbolization of the authority and personality of the individual who possessed the name. To us a name is far less important. It may be changed whenever one can convince a judge that another name might be more economically advantageous. The name is also a legal method of giving one’s written assent to certain business transactions, but to us it is not the symbol of the personality.”

The translation in Eastern Huasteca Nahuatl typically is “in someone’s authority” (for instance “I have come in my Father’s name” in John 5:43 becomes “I have come on my Father’s authority”) (source Nida 1947, p. 179), or in Highland Puebla Nahuatl with the more paraphrastic equivalent “as though on orders from you” or in Tzeltal as “by your authority, so he said” (both examples for Mark 9:38 and 39, see Bratcher / Nida).

In Guhu-Samane, Mark 11:9 (in English: “Blessed is he that comes in the name of the Lord”) is translated as “Blessed is the Lord’s namesake who comes.” “In the name of the Lord” caused “puzzlement [because] has he just assumed the name of the Lord, valid or otherwise? [But] with ‘blessed is the Lord’s namesake who comes’ the strong bond between the namesake and the important ancestor for whom named entitles the namesake to the deference due the ancestor. Thought very proper in this context.” (Source: Ernest L. Richert in Notes on Translation December 1963: p. 4-7; reprinted in The Bible Translator 1965, p. 198ff. )

Barclay Newman (see The Bible Translator 1974, pp, 432ff. ) reports on different solutions for the translation of the Today’s Malay Version (Alkitab Berita Baik, 1987):

In Malay “the phrase ‘in my name’ is problematic (…) since it sounds like the use of magic. For this reason [the English] Today’s English Version (Good News Bible) was followed at such passages as John 5:43 and 10:25, where ‘in the name of my Father’ is translated as ‘with my Father’s authority’ and ‘by my Father’s authority’ [respectively]. In John 12:13 ‘in the name of the Lord’ has become ‘in his (the Lord’s) behalf,’ following the common language German translation Die Gute Nachricht. In John 14:13, ‘because you are my followers’ is used, in John in 15:16, 16:23 and 24 ‘as my followers,’ in John 17:11 ‘by your own power, the power you gave me,’ and in John 14:26 ‘in my place.'”

Other translations for “in the name of Jesus Christ” include “in the authority of Jesus Christ” (Isthmus Mixe), “calling on Jesus Christ” (Teutila Cuicatec), “calling the name of Jesus Christ over you” (Ayutla Mixtec), “because of Jesus” (Tepeuxila Cuicatec), “by the power of the name of Jesus Christ” (Chichimeca-Jonaz), “the word of Jesus Christ is strong” (Lalana Chinantec) (source for this and above: Viola Waterhouse in Notes on Translation August 1966, p. 86ff.), and “mentioning the name of Jesus” (Elhomwe — source: project-specific translation notes in Paratext).

See also not speak in the name of Jesus, be baptized in the name of Jesus Christ, and as the LORD lives.