I'm sorry, but reading Rence immediately made me think of former White House Chief of Staff, Reince Priebus. (In his case, Reince is a nickname for Reinhold.) He's had the shortest tenure of any Chief of Staff in American history, as far as I know. And he's not a very popular guy.
The name Rance (pronounced like Lance but with an R) had some popularity in the U.S. in the 1800s, and BtN says it's likely derived from a Scottish surname. That's probably a better choice.
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