From a completely cynical perspective, the reason C-3PO has a red arm in Star Wars: The Force Awakens is to make it easier for Disney to sell a new wave of toys featuring the droid.
However, a new comic book from Marvel by James Robinson and Tony Harris offers up a very emotional and surprising explanation. It starts with Threepio and a group of Resistance droids cash landing on a planet with their captive, a First Order RA-7 protocol droid named Omri, who has information pertaining to the whereabouts of the captured Admiral Ackbar.
What this issue essentially does is humanise the droids by making us realise that they too live their own lives, lives which are stolen from them every time their minds are wiped by their human masters. Ultimately, Omri sacrifices himself to save C-3PO, and as acid rain pours down on him, his shiny exterior is washed away to reveal a past life where he was red.
An alien is responsible for tearing off Threepio’s arm, and after reflecting on his own past life, the droid decides to take the now dead Omri’s arm as a way of respecting his sacrifice.
It makes for a great read, but with the droid once again returning to his golden arm by the time Star Wars: The Force Awakens ended, not something we’re likely to see acknowledged on the big screen. Even so, getting some perspective behind this change is pretty cool.