The final piece of the minium facts approach details the conversion of Jesus’ unbelieving brother, James. Why is this such an important piece of evidence and why should we believe it’s true?
The conversion of the Apostle Paul is nothing short of miraculous, however, Paul wasn’t the only staunch non-believer who converted to the Christian faith following Christ’s death. Jesus had at least four brothers and among those was James. During Jesus’ life on earth, His brothers often mocked, challenged, and taunted Him. They never followed Him, worshipped Him, or spoke of His name and works with the Jews (see link 1 below. See also Mark 3:21, 31; 6:3–4; John 7:5). However, following Christ’s death, something happened that turned James from one who laughed to one who followed. What happened?
In order to grasp the significance of this conversion we should understand who, exactly, James was. If you’re like me, you might have imagined James, along with Jesus’ other brothers (Joseph, Judas, and Simon), to be something along the lines of delinquents or troublemakers. However, what is vital to note is that James (along with his brothers) was a pious Jewish believer who followed the law with conviction and honour. Eusebius, quoting from the now lost work of Hegesippus, says of James,
James, the brother of the Lord, succeeded to the government of the Church in conjunction with the apostles. He has been called the Just by all from the time of our Savior to the present day; for there were many that bore the name of James. He was holy from his mother’s womb; and he drank no wine nor strong drink, nor did he eat flesh. No razor came upon his head; he did not anoint himself with oil, and he did not use the [public] bath. He alone was permitted to enter into the holy place; for he wore not woolen but linen garments. And he was in the habit of entering alone into the temple, and was frequently found upon his knees begging forgiveness for the people, so that his knees became hard like those of a camel, in consequence of his constantly bending them in his worship of God, and asking forgiveness for the people. Because of his exceeding great justice he was called the Just, and Oblias, which signifies in Greek, ‘Bulwark of the people’ and ‘Justice,’ in accordance with what the prophets declare concerning him. 
In the same chapter, Eusebius, quoting Clement of Alexandria, describes James’ martyrdom,
Leading him into their midst they demanded of him that he should renounce faith in Christ in the presence of all the people. But, contrary to the opinion of all, with a clear voice, and with greater boldness than they had anticipated, he spoke out before the whole multitude and confessed that our Saviour and Lord Jesus is the Son of God. But they were unable to bear longer the testimony of the man who, on account of the excellence of ascetic virtue and of piety which he exhibited in his life, was esteemed by all as the most just of men, and consequently they slew him.
James was martyred for his belief in the risen Christ and this is attested to in both Christian and non-Christian sources. The Jewish historian Josephus notes that James was executed as a “breaker of the law”  the same way the Christians were (Acts 6:13; 18:13; 21:28).
Critics posit the argument that, following Jesus’ death, James could have been overcome with grief and guilt that resulted in his conversion. But it is important to ask if mere grief could sway one to abandon the belief, doctrine, and faith (i.e. pistis meaning loyalty) they once held. A belief that clearly states, Thou shalt have no other gods before me (Exodus 20:3). If critics are right, James would have to not care to verify that the disciple’s claims were true (and this, of course, assumes that the disciples really did claim that Jesus had been raised), which, in the context of their social world, would be inconceivable. Moreover, even if guilt and grief could make him reconsider his entire religion, a physical resurrection from the dead would have been the very last thing to cross his mind, if it were to cross it at all (see link 2 below). Yet, James is mentioned by name in our earliest source (1 Corinthians 15:3–7) and goes on to become the leader of the Jerusalem church (Acts 15:12-21; Gal. 1:19). Likewise, his brothers would also come to worship Jesus as God (Acts 1:14).
The account of this proud and pious Jewish believer who once mocked and taunted to one who followed and believed in his brother’s resurrection, even in the face of death, is nothing short of extraordinary. Like the conversion of Paul, it carries a weight that deserves to be feared. To this effect, scholars Hershel Shanks and Ben Witherington note,
“It appears that James, like Paul, was a convert to the Jesus movement because at some juncture he saw the risen Jesus, for nothing prior to Easter can explain his having become such a follower of Jesus, much less a leader of Jesus’ followers.”
Hershel Shanks and Ben Witherington, The Brother of Jesus: The Dramatic Story and Meaning of the First Archaeological Link to Jesus and His Family (London: Continuum, 2003), p. 107–9.