What does John 7:3 mean?

Verses 3-5 present both a sad commentary, and produce controversy. The men referenced are called Jesus' "brothers," a statement which conflicts with the idea that Mary, Jesus' mother, was a perpetual virgin. The Greek term is adelphoi, which can be used to refer to countrymen—meaning fellow Jews—as well as biological siblings. However, this same phrasing is used in verses such as Matthew 12:46, and Mark 6:3. In the gospel of Mark, Jesus' brothers are even named. Psalm 69:8 refers to "mother's sons," rather than "father's sons." This is the same passage which came to mind when Jesus cleansed the temple in John 2:16 (Psalm 69:9). In short, the only reasonable interpretation is that these were Jesus' actual, physical, half-brothers, and children of His mother, Mary.

The suggestion made here is that Jesus should go and put on a show in Jerusalem. The feeding of thousands, from chapter 6, attracted a great deal of attention. Given that these family members don't actually believe that Jesus is the Messiah (John 7:5), this is probably a sarcastic response to so many disciples abandoning Jesus after His preaching in Capernaum (John 6:66).

Tradition suggests that one of these brothers, James, would become a believer and write the New Testament book of James. Another brother is believed to have written the book of Jude. At least some of this disrespect might have been due to familiarity. Jewish tradition suggested that the Messiah would be relatively unknown (John 7:27), so merely knowing Jesus was enough for some to dismiss Him as the Promised One. As seen in the life of men like Judas Iscariot, simply knowing "about" Jesus, or being exposed to Him, is not enough to make a person a believer.