Paul was formerly an Other who joined the DHARMA Initiative. If Paul had betrayed the Others to marry Amy, it may also explain their motivation for killing him.
Ben punished Widmore by exiling him for having a child with an outsider. As Widmore was much more strict then Ben, it makes sense he would have the Others kill Paul for marrying Amy. Also explains why he had an ankh, a traditional Other symbol, and why they wanted his body back.
The above may be true, but in reverse. The necklace may have originally been Amy's and a gift to her husband. Amy may have been the Other originally.
Paul's name is a joking reference to Paul Anka. Similarly, the writers named two of the hippies after Jerry and Phil from the Grateful Dead.
Paul's last name is Rom. If Paul was an Other, Ethan could have chosen to take his mother's previous last name (one that had a connection to the Others) as a kind of Rite-of-Passage to becoming an Other himself after the Purge.