The phrase is definitively Johannine, occurring five times in four verses, all in Johannine texts. The Greek syntax, at a minimum, consists of a conjugation of the verb πιστεύω, the preposition εἰς, and the noun τὸ ὄνομα.1
But as many as received him, he gave power to them to become sons of God, [even] to those who believe in his name.
ὅσοι δὲ ἔλαβον αὐτόν ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ TR, 1550
Now when he was in Jerusalem during Passover, during the feast, many believed in his name when they saw the signs that he did.
Ὡς δὲ ἦν ἐν Ἱεροσολύμοις ἐν τῷ πάσχα ἐν τῇ ἑορτῇ πολλοὶ ἐπίστευσαν εἰς τὸ ὄνομα αὐτοῦ θεωροῦντες αὐτοῦ τὰ σημεῖα ἃ ἐποίει TR, 1550
He who believes in him is not judged, but he who does not believe is already judged, since he has not believed in the name of the only-begotten son of God.
ὁ πιστεύων εἰς αὐτὸν οὐ κρίνεται ὁ δὲ μὴ πιστεύων ἤδη κέκριται ὅτι μὴ πεπίστευκεν εἰς τὸ ὄνομα τοῦ μονογενοῦς υἱοῦ τοῦ θεοῦ TR, 1550
I wrote these things to you who believe in the name of the son of God, so that you know that you have eternal life, and so that you believe in the name of the son of God.
ταῦτα ἔγραψα ὑμῖν τοῖς πιστεύουσιν εἰς τὸ ὄνομα τοῦ υἱοῦ τοῦ Θεοῦ, ἵνα εἰδῆτε ὅτι ζωὴν ἔχετε αἰώνιον, καὶ ἵνα πιστεύητε εἰς τὸ ὄνομα τοῦ ὑιοῦ τοῦ Θεοῦ TR, 1550
According to Edwin Abbott Abbott in Johannine Vocabulary: A Comparison of the Words of the Fourth Gospel with Those of the Three,2 who cites Origen and John Chrysostom, "to believe in his name" (πιστεύειν εἰς τὸ ὄνομα αὐτοῦ) (i.e., of the Lord Jesus Christ) is distinct from "to believe in him" (πιστεύειν εἰς αὐτὸν). The former refers to the rudimentary faith (belief) expressed in baptism in the Lord Jesus Christ, while the latter refers to a more firm and secure faith (belief) in the Lord Jesus Christ.
Abbott, Edwin Abbott. Johannine Vocabulary: A Comparison of the Words of the Fourth Gospel with Those of the Three. London: Black, 1905.
1 John 1:12 and 2:23 follow τὸ ὄνομα ("the name") by αὐτοῦ ("his"), while John 3:18 follows it by τοῦ μονογενοῦς υἱοῦ τοῦ θεοῦ ("of the only-begotten son of God"), and 1 John 5:13 follows it by τοῦ ὑιοῦ τοῦ Θεοῦ ("of the son of God").
2 p. 34-37, §9, [1483-1487]