Words suffixed by '-onym' relate to different classifications of word, or more often, name.
Is there one?
It appears that I may have gone to EL&U too quickly; upon further inspection of a list of '-onym' words I found
Hypocoronym: A colloquial, usually unofficial, name of an entity; a pet-name or "nickname" (Wikipedia)
While this seems to fit the bill perfectly, it is the only definition I can find and is from Wikipedia. Since Wikipedia is not considered a reputable source for EL&U answer I have tried to find others, but they all seem to have the same text as the Wikipedia source.
Can anyone find a more reputable source?
The short answer is that there is none. There is no word ending in '-onym' that is equivalent to nickname.
But there are related terms that capture what 'nickname' captures. A nickname, which etymologically means an 'also' name, can be a diminutive of the original name, a street name, an AKA, fanciful or not. Given that, the possibilities are:
Some words that do not mean nickname but are close:
The closest to your specifications is cognomen. If I were to make up a word, it would be paranym (purely by literal root meanings) but that is already taken.
Not exactly the same as "nickname", a "pseudonym" is a fictitious name used by an author to conceal his or her identity. It's also called a "pen name"
pseudonym - "a fictitious name adopted, esp by an author" (from Greek pseudōnumon, neuter of pseudōnumos, falsely named : pseudēs, false; see pseudo- + onuma, name)
There are differences between a nickname and a pseudonym, though. While a "pseudonym" is usually chosen by somenone to conceal his identity, a nickname is usually chosen by friends, acquaintances or colleagues. In addition, a nickname can also be used for animals, cities, etc.