A diminutive (or pet name) of a given name is a short and/or affectionate form. Often they are only used by friends and relatives.
The most common diminutives (at least among English names) are those that are short forms of the original name, very often from the first syllable or sound of the name. For example, Alex is from Alexander, Barb is from Barbara, Deb is from Deborah, and Mike is from Michael. Other short forms don't come from the beginning of the name, but instead from the end or the middle, like Beth from Elizabeth, Fred from Alfred, Greta from Margareta, and Lisa from Elisabeth.
Suffixes
Diminutives can also be formed by adding a suffix to the original name or the name's short form. In English, the -y/-ie suffix is very common, leading to diminutives like Abby, Debbie, Charlie, Johnny and Sammy.
(From Behind the Name)
I'm glad we agree that nicknames are different @Mumtobabyhavoc. OP did update to say they weren't actually looking for a nickname.