I saw some web references that said that “Heinz” was just a variant of “Heinrich,” the German(ic) form of “Henry.” Accurate? Is “Heinze” just a variant of “Heinz”?
Yes.
At least it looks like that. However that is just the most obvious conclusion, I don’t know for sure. It is clearly not a common ordinary word in German. So far all the Google hits seem to use it as a surname.
Maybe “Heinze” is some shorthening of “Heinzerle” ? (never heard/saw this variation used as “nickname” pronounced or written as “Heinze”, but I am not German)
Salaam. A
If I remember correctly Heinz is a dialect word which denotes a large rack on which fish is dried.
If the word is neuter or masculine, Heinze could be an inflected form of that word, either plural, or one of those inflected singular datives kellner and I were talking about here.
It is also a given name. In my first high school German class, we had to choose German names for ourselves, and I chose that name.
“Heinz” would not be a nickname for “Heinzerle”, it would more likely be the other way around. “-erl” in some dialects is a diminutive prefix like the more common “-chen”, and is used with a name to denote fondness or that the person is actually of small stature, perhaps a child.
That is what I mean.
That “Heinzerle”, was for some reason shortened to “HeinzE” (because the OP mentions also HeinzE)
Not that Heinz is a nickname for Heinzerle instead of the other way round.
Heinzerle can be used when speaking of a child, but among friends also and then often as some sort of jokingly expressing you have a lot of sympathy for someone.
(I used to be very good in the dialect of Bayern. Sounded very genuine) 
Never heard about “Heinz” used for that fish drying thing you mention. Do you know which dialect that is?
Salaam. A
I’m almost positive that Heinze is an inflected form. It’d be either any plural case except dative, or the obsolescent singular dative. In making this statement, I’m assuming that it’s a masculine or neuter noun rather than a feminine one.
Other languages with diminutive endings seem to use them in a similar way. In Spanish, I’ve seen the use of -ito, -ita endings that work much the same way.
I saw the word Heinz in my Cassell’s German/English Dictionary. I don’t remember the edition, but it would have been in the late 1970s when I was a student. I don’t think it indicated a dialect, but if I can dig out the book again I’ll check.
Obviously fish-drying is not something that is done by 99% of Germans, so it’s understandable that even native speakers might not be familiar with the word.
By the way, I’m sure everyone knew I meant to say suffix, not prefix.