Cox is one of Australian domestic cricket's most-decorated players, having played the most Pura Cup-Sheffield Shield games in the competition's history (161).
He is the second-highest run-scorer in Sheffield Shield history with 10,821 runs, second only to South Australian captain Darren Lehmann.
Cox was a selector for his county side Somerset for four years in England.
"I'm sure there will be some challenging times ahead and some tough decisions to be made, however, that comes with the role and is something that any selector will experience," Cox said in a CA statement.
"We have a fantastic core of experienced players along with a talented and ambitious group of young players who are emerging into international cricket.
"There will certainly be some transition in the side over the next few years so it is an exciting time to be involved with the group."
Cox retired from first-class cricket in 2005-06, with his career spanning 264 first-class matches over 19 years for Tasmania and Somerset.
Border quit as a selector last month amid reports of a conflict between beer sponsors.
Border, who played 156 Tests and 273 one-day internationals and serves on the Cricket Australia board, was a selector between 1998 and 2005 before rejoining in June this year.