Yes and no.
Nazi can be and was a diminutive form of the name Ignatius. In Bavaria and Austria is is no longer that popular in this form now.
To name a quite prominent example:
Johann Nepomuk Eduard Ambrosius Nestroy (7 December 1801 – 25 May 1862) was a singer, actor and playwright in the popular Austrian tradition of the Biedermeier period and its immediate aftermath.
Out of the 83 recorded works of Nestroy, a total of some 56 were designated as some form of Posse, meaning a farce or 'broad comedy', including 32 Possen mit Gesang (farce with singing).
In his play Eulenspiegel (Nestroy, 1835) we find a figure called Natzi in the German version (PDF [here] (http://daten.digitale-sammlungen.de/~db/0005/bsb00050466/images/)4) or sometimes OCRed as Nazi in the Austrian version (archive.org). (Full text online on zeno )
One of his first lines in that play is
NATZI. So einen Eigensinn, wie die hat, den find't man nicht in alle fünf Weltteil', nicht in Europa, Amerika, Asina, Afrika und Paprika.
__transl__You can't find such a stubbornness in all five parts of the world, not in Europe, America, Asina, Africa and peppers.
Then shortly after
NATZI with piquancy to Mehlwurm. If we two stand next to each other, it is just like two flour sacks'; one is the reject, the other is the extract. With reference to Mehlwurm. The reject is the rough one, there is therefore little demand', that remains behind. With relation to itself. The extract is the best, it is sought, it is strong, that's why I go. Come d' Mrs. Mother! Goes off with Cordula to the middle.
We might read a lot of foreshadowing into that.
Anyway, there is quite some prior art for using Nazi to refer to a simpleton:
Google Books gives the examples of
Nazi Semmelbachers hochzeitsreise, 1911
Auerbach's Dorfgeschichten, 1848
Most convincingly this pars-pro-toto usage is illustrated in
Arthur Michelis: Reiseschule für Touristen und Curgäste. Leipzig, 1869.
Denn das Heu, das seit Monaten einen Bretterverschlag gefüllt und irgend einem Nazi oder Sepperl zur Lagerstätte gedient, hat einen andern Athem, als das, Großstädtern bekannte, frisch gemähte, auf einer Wiese ausgebreitete.
__transl__For the hay that has filled a wooden shack for months and served as a deposit for some Nazi (Ignatz) or Sepperl (Joseph) has a different breath (air) than the freshly mown hay known to city dwellers spread out on a meadow.
The first use of Nazi to insult someone is older than the Hitler party. When they presented the name and ideas, that contraction 'came natural'. And it was partly embraced early on by party members:
"The Nazi-Sozi" by Joseph Goebbels, 1926 (Ten Commandments for Each National Socialist).
In 1922 Kurt Tucholsky then closes the circle to illustrate that Austrian provincialism and chauvinism is closely connected to the term:
I am neither a relative nor a relative of the accused (we would both strongly forbid it) - but as far as the immigration of foreign elements is concerned, there is one case where I liked to climb onto the Berlin City Hall and, looking at the capital lying at my feet with tears fluttering around my eyes, would say: "You shouldn't have it! Namely the "Nazis" not Berlin.
Don't look at me, reader. Let us quickly agree that by the "Nazis," I mean that certain genre of the Austrian, Moravian, and especially Viennese artist folk, which begins to pollute the above-mentioned Berlin in the most violent way. But we don't want that any more.
I know exactly what an unfortunate role the Poznan-born Berliner plays on journeys ("Det is der Kölner Dom! Haben Se keenen größern?"). But no Berliner has ever been as cheeky and local-chauvinistic as this kind of "Nazis" (which I deliberately don't call Austrian, because Otto Weininger is one and Peter Altenberg and Karl Kraus and Alfred Polgar - but we agree who we mean).
The University of Vechta has a FAQ-like tidbit on the development:
The Illustrierte Lexikon der deutschen Umgangssprache by Heinz Küpper (Stuttgart 1984) states - in analogy with the Historical Dictionary of German Figurative Use by Keith Spalding (Oxford 1984) - "The abbreviation 'Nazi' referred in 1903 to the 'Nationalsozialen' under Friedrich Naumann. For the National Socialist first (?) documented by Kurt Tucholsky in 1923. In any case, Nazi propaganda minister Dr. Joseph Goebbels is not the creator" (p. 2021). –– The first known use of the word National Socialist is, by the way, even older; Cornelia Berning ("Vom 'Abstammungsnachweis' zum 'Zuchtwart': Vokabular des Nationalsozialismus", Berlin 1964, p. 138) she proves it for 1887 in the German Adelsblatt. Under the heading "Fürst Bismarck der erste Nationalsozialist" (Prince Bismarck the First National Socialist), it says: "The state is not, according to today's party system, a sum of individual wills, but the overall will as an expression of the National Spirit. Therefore, however, he knows, as only One God, also for the present only one reasonable party concept, namely, so one could say, National Socialism with the One Program of the Christian commandment of justice and love. The first representative of such a Unity National Party can be recognized in Prince Bismarck".
(Own translation, links added)
If the liberal Naumann raises concerns about word usage and proper history, perhaps National Socialism Before Nazism: Friedrich Naumann and Theodor Fritsch, 1890-1914 might help to better understand the word itself (but surely not the meaning of it in political analysis!).
(For completeness sake: the right-wing etymological conspiracy theory sometimes found on the web that the term nazi was coined by ancient Greek Jews is … complete confabulation relying on purposely misreading Nazirite.)
A related post on GermanSE https://german.stackexchange.com/q/32721/29694 reveals the widespread usage of Nazi in contrast to Sozi in Austrian newspaper from 1920 (example, example). But is seems that Nazi in these articles is of two almost different mothers: not really referring to German NSDAP members in the first but illustrating again the overlap of austro-fascism with provincialist chauvinism in Austria; in the second case describing tendencies in politics in Berlin. Before 1920 the term in Austrian newspapers is apparently reserved as a nickname for actual persons and as the character stereotype. (Most comics would have been overdoing it as "Nazis", src, 1903)
The above should prove sufficiently that the following summary is roughly correct:
The use of the term Nazi is a well-established convention in English. It emerged around 1924 among opponents of National Socialism, who borrowed it from the nickname Nazi (from the masculine proper name Ignatz, the German form of Ignatius), meaning “a foolish, clumsy, or awkward person.” The NSDAP briefly adopted the Nazi designation as what the Germans call a “spite word,” but they soon gave this up and generally avoided the term, considering it derogatory. Before 1930, English speakers had called the party members National Socialists, a term that dates from 1923. The use of “Nazi Germany,” “Nazi regime,” and so on was popularized by German exiles abroad. From them, the term spread into other languages and eventually was brought back to Germany after the war.
Anson Rabinbach & Sander L. Gilman (Eds): "The Third
Reich Sourcebook", University of California Press: Berkeley, Los Angeles, London, 2013, p 4.