A 5 second Google (is your friend) shows the operation referred to as digit sum or sum of digits. WA even has an operator that performs this.
Tangentially related, you might also look at the concept of the checksum. I'm familiar with that from military navigation/targeting, but the wiki article discusses a lot of other uses. It's tangential, I guess since, per wiki, the checksum is not forced down recursively to a single digit. (Although I've seen some military use where it was, because the field had a defined digit size of 1.)