In the Synoptic gospels you have Joseph of Arimathea being given the body of the dead Christ. Why would the body be given to a stranger and not back to Jesus's family? What do we know of the rights of the deceased in first century Roman rule?

In his book How Jesus Became God, Bart Ehrman makes the case that it is unlikely that Jesus was buried by Joseph of Arimathea, or in fact that he was buried in a private grave at all.

So, a little background. According to the gospel of Mark, Jesus was buried by a previously unnamed character, Joseph of Arimathea, a member of the Sanhedrin. Mark 15:43 says "Joseph of Arimathea, a respected member of the council... took courage and went to Pilate and asked for the body of Jesus." Joseph then takes the body from the cross and buries it "in a tomb that had been hewn out of the rock," and rolls a stone in front of the tomb.

Here's why this is unlikely from a historical perspective:

Earlier in Mark's gospel, the whole Sanhedrin was said to look for evidence "against Jesus to put him to death" (14:55), and at the end of the trial, "they all condemned him as deserving death" (14:64). Why would Joseph, a member of the Sanhedrin, then take it upon himself to ask Pilate permission to give Jesus a proper burial?
In our earliest record of the resurrection (the writings of Paul), he simply says "he was buried" without mentioning Joseph of Arimathea (1 Corinthians 15:3-8), even though that would have made sense given the literary structure of the statement.
Also in 1 Corinthians 15:3-8, Paul seems to make an exhaustive list of all the witnesses to the resurrected Jesus, without mentioning the women who supposedly first saw him according to the gospel narratives.
In Acts 13, in one of Paul's speeches (likely composed from previous oral traditions), he says, "Though they could charge him with nothing deserving death, yet they asked Pilate to have him killed. And when they had fulfilled all that was written of him, they took hi down from the tree and laid him in a tomb" (Acts 13:28-29). Here it is suggested that the council as a whole buried Jesus, not a single member "Joseph." Ehrman suggests this may be an earlier tradition than the Joseph of Arimathea tradition.
There is lots of evidence that normal Roman practice was to leave the crucified person's body on the cross to decompose and be eaten by scavenging animals, and/or be thrown into a mass common grave. This was seen as part of the punishment, to not have a proper burial.
Furthermore, exceptions were only made in very rare cases.
Pilate, the Roman governor, was known to be a ruthless and harsh ruler with no mercy whatsoever for Jewish sensibilities.

Ehrman also points out:

Let me stress that all of this--or something very much like it--needs to happen within Mark's narrative in order for what happens next to make sense, namely, that on the day after the Sabbath, Mary Magdalene and two other women go to the tomb and find it empty. If there were no tomb for Jesus, or if no one knew where the tomb was, the bodily resurrection could not be proclaimed. You have to have a known tomb.

How did Jesus's followers become convinced that Jesus was raised from the dead? The most likely answer seems to be that it was from their visions of him. After all, even in the gospel narratives, the people who discover the empty tomb do not assume that he was raised from the dead until they actually see him!

There is a lot more detail in the book, and it's a highly interesting read. I'd recommend it.

EDIT: In response to your question about why the body would be (hypothetically) given to a stranger and not family, Ehrman notes that in all the stories Jesus's family was miles away in Galilee. Not only that, but they were poor and would have had no resources for a proper grave in Judaea. In addition, the earliest accounts state that the disciples all fled the crucifixion scene for Galilee, leaving no one to bury Jesus but the unlikely Sanhedrin themselves.