Kim,
our first elf was Skeeter Jingle Peppermints, he came in 2008 and he already had a name, that same year I had gotten an elf for my nephew and his sisters (Skeeter's twin sister, Penelope) and also got an elf for my mom, my middle son (who was 6 at the time) spotted my mom's elf at a store and said Mom, they are fake I just saw a bunch of elves that look just like Emerald, so I had to quick think of a story and told him that sometimes Santa sends his magic elves to the stores for family's who don't know about magic elves, to buy, he insisted on buying one to see if it was magic, and he prayed and prayed for it to be a magic elf, then he wrote a letter to Santa just to check and see if his elf could be magic, so Santa wrote back and said that the elf would need to come to the north pole to complete the EIT program., When the elves left for EIT (my dd had also by then gotten one) they did not have names, so Santa wrote and asked them if they would like to help name the elves.
now they each have their own
You could make up adoption papers for your elf/elves and leave the name blank for the kids to put in, like how cabage patch doll adoption papers used to be