Both and neither!
Elvis’ middle name was to honor his father’s friend, Aaron Kennedy. But the Presley’s (Vernon and Gladys) chose the spelling Aron. His stillborn twin brother Jesse was named to honor Vernon’s father, Jessie Presley, but again with a slightly different spelling.
In his later years, Elvis wanted to change the spelling of his middle name to the more biblical Aaron. He was surprised to find that unlike his birth records, the official state records already had it as Aaron.
Knowing Elvis’ preference for Aaron, that is the name his father ordered to be placed on Elvis’ tombstone. And it is the officially recognized spelling by the Presley estate today.
It is unclear why the marker at Graceland for Elvis’ brother is spelled Jessie. The baby was buried in an unmarked grave in Tupelo because the family was poor. Today the Presley estate uses Jessie as the official spelling.