In my lifetime, the two most common words used to describe names other than the family name are given names and Christian names. The latter has been steadily falling out of favor. Indeed, a google ngram view of the two shows exactly that phenomenon: n-gram of christian name, given name. Although, it should be noted, in common usage both labels are applied to mean the first name, or commonly used name (should it be other than the first) only. There is no single word, in English, that will automatically be understood by the recipient of a communication (verbal or written) to mean "all the names a person has not including their family name".
(Edit: for an interesting comparison, add forename to that n-gram view.)