So the title says it all really. The term hamburger doesn't refer to ham but instead the origin of the food Hamburg, but when the presence of cheese was added the new invention is referred to as a cheeseburger.
What is this type of derivation called. Where part of a word is changed due to a misunderstanding (or other reason) of a part of an original.
I recall being told this fact, but I dont remember the name for this type of derivation
This is called rebracketing: when the original [hamburg][er] is reinterpreted as [ham][burger]. Other examples include [alcohol][ic] > [alco][holic] and [helico][pter] > [heli][copter].
It is unclear whether cheeseburger was actually formed based on a misunderstanding of the etymology of the word hamburger. It can be noted that the word burger without the ham is also in frequent use. Instead it could be argued that -burger has become a libfix:
In linguistics, a libfix is a productive bound morpheme affix created by rebracketing and back-formation, often a generalization of a component of a blended or portmanteau word.
(Wikipedia)
Some examples mentioned in other answers and comments, such as -copter, -holic and -gate, can also be considered to be libfixes.
There is some folk etymology involved: First, hamburger is given a folk etymology containing actually ham and then new words can be formed after the perceived model of hamburger.
The term rebracketing mentioned in the answer by Draconis is more general, it also covers cases where no folk etymology plays a role, like Sandhi shift (a napron -> an apron; an adder -> a nadder).
EDIT: And there is another term to watch: -burger is called a pseudo-suffix.
Two relevant terms are back-formation and reanalysis (along with rebracketing and folk etymology) already mentioned in other answers).
Reanalysis is (in my experience) the most general of these, covering all cases where a surface form originally built up in one way (morphologically, phonetically, or otherwise) is later treated as though it had been built up in a different way: e.g. hamburg-er to ham-burger, or a napron to an apron.
Back-formation is a bit more specific, typically applied to cases where a a new word is caused by removing affixes from a reanalysed compound; so e.g. an apron would not usually be described as a back-formation, burger from hamburger might be, but pea from pease (originally a mass noun, not a plural) certainly is.
a change in a word or phrase resulting from the replacement of an unfamiliar form by a more familiar one.
In etymology, back-formation is the process or result of creating a new word via inflection, typically by removing or substituting actual or supposed affixes from a lexical item, in a way that expands the number of lexemes associated with the corresponding root word
E.g. for "resurrect" to exist "resurrection" had to be reanalyzed as *verb + -ion, or /ʃn̩/ as the case may be, which happened on a wider scale with similar words. Likewise, ham and other words in the same category likely combine with many words, so it looks like a productive pattern. That's analogy when entire paradigms coalesce. This is not really the case here, but it is not a closed class either.
By the time that Cheeseburger was coined, the Hamburger was likely associated with ham for a long time, though it would be a peculiar name when paddies are made from beef. The fact that burger had hitherto no meaning on its own has no bearing, because regular words like bread, loaf or cran as cranberry-morpheme too have no internal derivation to justify their use.
Linguistic relativity is narrower than that. The same holds in principle for ham. We don't have to expect an initial etymology passed on with the condiment, because Hamburg has no monopoly on meat in a bun. To expect users to conform to some lexicographic ideal is the etymological fallacy
Since Mett-Brötchen (cp. minced meat) are popular in northern Germany, and Hamburg's markets are popular for their fresh food, raw ham is a likely ingredient. It's possible that sailors needed conserved meat balls, but there is no obvious significance to the receipt other than perhaps clever small time marketing, which cannot be excluded. Whether in a deliberate pun or by accident, whether from ham ("pork"), hum (a big bite), archaizing ham ("home") ... if the word ham contributed to the proliferation of hamburger as a word, that's certainly notable from a German perspective where ham is otherwise less meaningful. It is not even wrong since the details of its origin are admittedly a bit hazy.
That would make the Hamburg origin story a folk etymology, in my humble opinion, that is, a notable account which is probably, to a certain extend, correct yet inaccurate. The older a folk etymology is, the more likely it is either still uncertain or eventually incorrect. 1. If they are correct they are simply called etymology, or specifically transparent. 2. If the origin is obscure, that's why a folk etymology emerged to stop the gap in the first place. 3. Etymology as an ongoing effort has made advances that turn recent etymologies (1) and long standing riddles (2) obsolete. A particular subset of these is the folk etymology as an object of lexicographical interest. Even if it is obviously wrong, its origin might be informative at the root.
That's why cheeseburger cannot be said to be based on a folk etymology: ham was not replaced with a more familiar ham, its meaning was rather extended one way or another, and it was not replaced with cheese for reasons of familiarity. It's a recent economic coinage.
https://en.wiktionary.org/wiki/Folk_etymology
https://en.wiktionary.org/wiki/Back-formation
