Where does the Bible tell us the name of the “beloved disciple”?

How this might disprove sola scriptura is a mystery.

But for a fact, Stephen Ray's vaunted Teaching Authority hasn't infallibly provided the answer either...

Clearly waiting on them is not wise....how many more centuries will they need?

Sola Scripturists can deduce who this is from scripture, and even take into account any relevant tradition:

In three passages, however, an unnamed friend of Peter is described as “the disciple whom Jesus loved.” First, the Beloved Disciple has a place next Jesus at the Last Supper, and Peter beckons to him to discover the name of the traitor. This must have been one of the Twelve (see on 13:23), and so his identification with John the son of Zebedee is suggested.

Secondly, Peter and “the other disciple whom Jesus loved,” run together to the sepulchre which Mary Magdalene had reported to be empty (20:2f.). The Beloved Disciple’s eagerness to be first at the tomb, his hesitation to enter it when it was reached, and his “belief” when he saw that it was empty, are graphically described.

Thirdly, the two disciples whose fates are contrasted in 21:15–23 are, again, Peter and ὁ μαθητὴς ὃν ἠγάπα ὁ Ἰησοῦς; and the latter is, apparently, a fisherman, as we know John the son of Zebedee to have been. The narrative of the Appendix helps the identification in another way. The “Beloved Disciple” must be one of the seven persons indicated in 21:2, and among these the sons of Zebedee are expressly included. James is excluded, for the tradition of v. 23 could not have arisen in regard to him (Acts 12:2), so that if the Beloved Disciple were not John the apostle, he must be either Thomas, Nathanael, or one of the two innominati (see on 21:2 for the possibilities).

Now the constant tradition of the early Church was that the name of the Beloved Disciple was John. Irenæus (Hær. iii. 1, 1) and Polycrates (see p. 1. below) are explicit about this. So are the second-century Acts of John (ἀνακείμενον ἐμὲ ἐπὶ τὰ ἰδια στήθη ἐδέχετο, § 89). So is Origen (cf. Eusebius, H.E. vi. 25). This is a point on which tradition could not have gone astray, and there is no other tradition. There can be no reasonable doubt that the name of the Beloved Disciple was John, and therefore Thomas and Nathanael are excluded.1 If there was another John among the two innominati, we might claim him as the Beloved Disciple, but for this there is no evidence.

The only other mention of the Beloved Disciple in Jn. is at 19:26, where he is standing near the Cross in company with the Virgin Mother, whom he received εἰς τὰ ἴδια “to his lodging.” This (see on 19:27) is not inconsistent with his being the “witness” to whom appeal is made in 19:35, for ample time had elapsed to permit of his return to the Cross. And when we find at 21:24 that it is the Beloved Disciple who is designated as “the disciple who bears witness of these things,” it is difficult to avoid the conclusion that the “witness” of 19:35 is the same person (cf. p. lxix below).- Bernard, J. H. (1929). A critical and exegetical commentary on the Gospel according to St. John.(New York: C. Scribner' Sons.)

__________________
The times of this ignorance God winked at; but now commandeth all men every where to repent:

Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me.-kjv