What does it mean when the Bible says, “For even his brothers did not believe in him?”
“Brothers” in Greek (adelphe) has a very wide range of meaning, including cousins, countrymen, uncles, etc. Catholics believe that Jesus’ “brothers” were, accordingly, cousins or perhaps half-brothers (from a previous marriage of St. Joseph), and, of course, that Mary was a perpetual virgin. The Bible never refers to Mary as anyone’s “mother” besides Jesus (see Mt 1:18; 12:46; 13:55; Mk 3:31; Lk 8:19; Jn 2:1, 3, 5, 12; 19:25-26; Acts 1:14).
Mark 3:21-22 (RSV) And when his family heard it, they went out to seize him, for people were saying, “He is beside himself.”  And the scribes who came down from Jerusalem said, “He is possessed by Be-el’zebul, and by the prince of demons he casts out the demons.” (cf. Jn 10:20-21)
Note the italicized and bolded word. Other translations (including, unfortunately, KJV, NKJV, NIV, NASB) make it sound like Jesus’ family were agreeing and/or saying that Jesus’ was mad, but in fact the text is saying that “people” in general were doing so (just as the Pharisees did).
But if the text doesn’t refer to them, it can simply be construed as his family coming out to remove him from the crowds, who were massively misunderstanding him, accusing, and perhaps becoming violent (as at Nazareth, when they tried to throw him over a cliff).
Hence, there would be no necessary implication of his family’s disbelief in him. They were concerned for his safety. Other translations convey the true sense of the passage (which is interpreted by 3:22 indicating that the “scribes” were saying Jesus was crazy):
NRSV When his family heard it, they went out to restrain him, for people were saying, “He has gone out of his mind.”
Good News / (TEV) When his family heard about it, they set out to take charge of him, because people were saying, “He’s gone mad!”
Moffatt . . . . . . for men were saying, “He is out of his mind.”
Phillips . . . for people were saying, “He must be mad!”
NEB . . . for people were saying that he was out of his mind.
But I grant that it’s certainly possible that some of Jesus’ relatives — thinking with the carnal mind that virtually everyone possessed before Pentecost (including the disciples, almost all of the time) — may have vastly misunderstood him. If so, nothing in that contradicts what Catholics believe.
We need only look at Peter (rather remarkably) rebuking Jesus (Mk 8:31-3; cf. Mt 16:22), to see that. We also know from Jesus that the prophet is without honor in his hometown (Mk 6:4; cf. Mt 13:57). So such a thing is not out of the question or utterly ruled out. But I have provided some arguments showing that it is not necessarily the case, from the text.
John 7:5 For even his brothers did not believe in him.
See the above reasoning! But again, there are some questions about exactly what this means, to “believe” in Jesus (before Pentecost). It could merely mean that they didn’t believe he was performing miracles, or did, but did not believe or understand — like most of the disciples, most of the time — that he was the Messiah (or even claimed to be). The Protestant Barnes’ Notes on the Bible has observations along these lines:
It appears from this that they did not really believe that he performed miracles; or, if they did believe it, they did not suppose that he was the Christ. Yet it seems hardly credible that they could suppose that his miracles were real, and yet not admit that he was the Messiah. Besides, there is no evidence that these relatives had been present at any of his miracles, and all that they knew of them might have been from report.
Expositors Greek Testament adds:
[T]his does not mean that they did not believe he wrought miracles, but that they had not submitted to his claim to be Messiah. They required to see him publicly acknowledged before they could believe.
The context of the preceding verse supports the latter take: “If you do these things, show yourself to the world.” They were perhaps thinking that he might be the Messiah; if so, he should go to Jerusalem and proclaim it and make a mighty show of attesting miracles (according to the then prevailing Jewish notion of the appearance of the Messiah). Until then, they remained skeptical. It seems perfectly plausible to me. Dom Bernard Orchard’s Catholic Commentary (1953) pursues this line of thought:
4. The brethren emphasize the apparent contradiction of working miracles and thereby wishing to be a public personage in obscure Galilee. They want him to show himself to the great world in the center of Judaism.
5. They had but an imperfect idea of his Messianic mission, since he was bringing no worldly glory to himself and them.
6. Christ’s answer means that the right time for a public ascent to Jerusalem involving a triumphant manifestation had not yet come — there were yet six months to Palm Sunday. The right time for the brethren is any time.
7. They have the peace of the worldly with the world; not so Jesus who has the hatred of the world for condemning its badness. . . . The appointed time for the public encounter with the full hatred of Jewry has not come.
If this is correct, again, it is not so much a manifestation of obstinate, stiff-necked unbelief (as with the scribes and certain of the Pharisees), but carnally minded, mistaken, incomplete, confused semi-belief, as with most of the 12 disciples at most times before Pentecost.
And this occurred right after John 6, where it is reported that “many of his disciples [i.e., more than just the twelve] drew back and no longer went about with him” (John 6:66) because they couldn’t accept the eucharistic Real Presence and transubstantiation that Jesus had been emphatically stressing, to their disdain (“This is a hard saying; who can listen to it?”: 6:60).