As evidenced by this historical footage, their most famous slogan was, after all, don't be stupid, be a smarty, come and join the Nazi party.

But in all seriousness, the term "Nazi" for members of the NSDAP and adherents to National Socialism was first coined by their opponents. "Nazi" previous to that was mostly in use as a shortening of the German name Ignaz and implied in popular perception a bumbling idiot, see e.g. the 19th century operetta "The Happy Nazi". When the term was first used is a bit unclear with some sources pointing to Kurt Tucholsky first using it in 1923 to describe the actual Nazis, other sources pointing to a Bavarian journalist first using the term the same year.

The actual Nazis had at first an ambivalent relationship to the term. Seen as an equivalent to the shortening "Sozi" for members of the socialist party, there were some attempts to reclaim the term in official literature, such as in Goebbels 1927 book "Der Nazi Sozi", which was his attempt to create a sort of FAQ about Nazism at a time when they didn't do well at the polls (also, a book in which Goebbels emphatically rejects the NSDAP as a left-wing party to use the actual Nazis to dispel a myth so popular in the US).

Beginning with their take-over of power, official literature began to avoid the term because of its initial derogatory tone. However, it still found some use in the press as a search in the Austrian national library's newspaper archive shows and the term caught on in popular use with scores of people, even members of the NSDAP, using it as a descriptor in their diaries for example as Janosch Steuwer showed in his recent book »Ein Drittes Reich, wie ich es auffasse« Politik, Gesellschaft und privates Leben in Tagebüchern 1933-1939 about the perception of Nazi Germany in German diaries.

This was a bit to the chagrin of the Nazi leadership for a variety of reasons:

First, Fascists have a persecution complex inherit in their ideology. Hence, they can see history as e.g. the "confrontation between Jews and Aryans". This makes them extremely sensitive to ridicule and humor (I think the reddit term would be "special snowflakes"). When somebody called out the ridiculousness of their aesthetic with the uniforms and the torches and the creating swastikas by marching in lockstep, they tended to react very violently. One such example that perfectly illustrates this is the number of Gestapo persecution cases during the war against people who had called their local NSDAP representative – who wore a ridiculous golden uniform – "gold pheasants". Basically, Fascists are very sensitive and their whole "we are so serious because we are a serious political movement" shtick is dispelled by ridicule and humor.

Second, names are important because they come to define how the public perceives one's movement and one's agenda. Calling themselves "National Socialists" rather than say "The anti-Semitic Party" (which had previously existed) was important to project to a serious image of their political agenda and in order to define themselves and define them in the eyes of others. This is a general lesson of politics. To use an example to illustrate this: It's basically the same reason why today there are people who call themselves "Men who go their own way", "Men's Rights Activists" or "The Red Pill" instead of laying out their agenda in the open and calling themselves "violent misogynists", "rabid sexists" and "bigoted men who like to beat women". Or why people use dogwhistles like "cultural Marxism" for their anti-Semitic conspiracy theories. The name is important because a whole slew of associations can be projected into it. Hence, also "The Third Reich" because it suggest continuity to the Holy Roman Empire and the German Empire instead of just "Germany". It's generally easier to be taken seriously when the designated term for one's movement isn't a term usually used for a bumbling idiot called "Ignaz".

So, to sum up: Originating with opponents of Nazism, the Nazis at first tired to reclaim the term but after their take-over of power gave up on that and purged it from official texts and literature. However, the term caught on in popular usage and they never managed to get rid of it.