The name of the ministry comes from Joseph of Arimathea, who appears in all four of the Gospels.
When it was already evening, since it was the day of preparation, the day before the Sabbath, Joseph of Arimathea, a distinguished member of the council, who was himself awaiting the Kingdom of God, came and courageously went to Pilate and asked for the body of Jesus. Pilate was amazed that he was already dead. He summoned the centurion and asked him if Jesus had already died. And when he learned of it from the centurion, he gave the body to Joseph. Having brought a linen cloth, he took him down, wrapped him in the linen cloth and laid him in a tomb that had been hewn out of rock. Then he rolled a stone against the entrance of the tomb. Mary Magdalene and Mary, the mother of Jesus, watched where he laid. Mark 15, 42-47.
At each funeral or memorial Mass, there are at least two Arimatheans who perform the following tasks: