When I thought of this, I made a list of all the blood relatives I could think of, from every team. I tried to do it without looking any of them up, unless it was to verify if someone ever played in a major league game. To qualify, you only need to have played in one.
Then I realized that if I put up the whole list it would be kind of hogging the thread. So instead I'll just give my list that includes at least one Red Sox player.
Fathers and sons, and brothers, obviously. We also have to include grandfathers and sons, for the sake of a certain Red Sox Hall of Famer. Cousins? I think there have been a few, but I couldn't think of them. Anyway, sure, why not.
Dom DiMaggio - Joe DiMaggio (HOF)
Ken Brett - George Brett (HOF)
Carl Yastrzemski (HOF) - Mike Yastrzemski
Tony Conigliaro - Billy Conigliaro - played on team at same time
Dante Bichette - Bo Bichette
Pedro Martinez (HOF) - Ramon Martinez - played on team at same time
Jeremy Giambi - Jason Giambi
JD Drew - Stephen Drew - both played on Red Sox
Daniel Bard - Luke Bard
One I had to look up was Papelbon. I wasn't sure if Josh ever pitched in the majors. I found out he didn't. I also found out there was another brother, Jeremy who pitched in the minors.