Well I'm surprised that no one has come forward here. The Nazis generally generate a lot of answers, but looking through the FAQ, there doesn't seem to be anything there.

So to start, I think its important to remember that Hitler was not so much the originator of the various racial theories embraced by Nazism as the inheritor of them. The specific origin of the term "aryan" to mean a subset of white, Christian Europeans who generally spoke a Germanic language is usually attributed to Arthur de Gobineau. Gobineau was a French aristocrat during the Third Republic and in his essays argued that the dominant and only creative branch of human society was a group called the Aryans, which he could trace from ancient times to the present day, primarily through linguistics with a whole lot of bad historical method mixed in. Because I don't think I can give it any better justice than the man himself, I'm going to quote at length from his essay The Inequality of Human Races. The following is a list of the great civilizations of world history:

I. The Indian civilization, which reached its highest point round the Indian Ocean, and in the north and east of the Indian Continent, south-east of the Brahmaputra. It arose from a branch of a white people, the Aryans.
II. The Egyptians, round whom collected the Ethiopians, the Nubians, and a few smaller peoples to the west of the oasis of Ammon. This society was created by an Aryan colony from India, that settled in the upper valley of the Nile.
III. The Assyrians, with whom may be classed the Jews, the Phoenicians, the Lydians, the Carthaginians, and the Hymiarites. They owed their civilizing qualities to the great white invasions which may be grouped under the name of the descendants of Shem and Ham. The Zoroastrian Iranians who ruled part of Central Asia under the names of Medes, Persians, and Bactrians, were a branch of the Aryan family.
IV. The Greeks, who came from the same Aryan stock, as modified by Semitic elements.
V. The Chinese civilization, arising from a cause similar to that operating in Egypt. An Aryan colony from India brought the light of civilization to China also. Instead however of becoming mixed with black peoples, as on the Nile, the colony became absorbed in Malay and yellow races, and was reinforced, from the north-west, by a fair number of white elements, equally Aryan but no longer Hindu.
VI. The ancient civilization of the Italian peninsula, the cradle of Roman culture. This was produced by a mixture of Celts, Iberians, Aryans, and Semites.
VII. The Germanic races, which in the fifth century transformed the Western mind. These were Aryans.
VIII. -X. The three civilizations of America, the Alleghanian, the Mexican, and the Peruvian.

Of the first seven civilizations, which are those of the Old World, six belong, at least in part, to the Aryan race, and the seventh, that of Assyria, owes to this race the Iranian Renaissance, which is, historically, its best title to fame. Almost the whole of the Continent of Europe is inhabited at the present time by groups of which the basis is white, but in which the non-Aryan elements are the most numerous. There is no true civilization, among the European peoples, where the Aryan branch is not predominant.

The term Aryan, by the way, was understood at the time (and still is to a certain extent today, I believe, although I'm not up to date on that literature) as signifying a group of conquerors in Ancient India who eventually installed themselves at the top of the then nascent caste system. I'm not an expert on Indian history, so I'm not sure how accurate that understanding is, but it's definitely accurate in the sense of that being what folks thought in the 19th and early 20th centuries. Gobineau does not state explicitly why he chose the term aryan and if you think about it, from his point of view, he didn't choose it, so much as he was simply recounting what he viewed to be the immutable truth. We can surmise, however, that he chose it because of the contemporary belief that this group was a ruling caste in the most ancient of the greater civilizations (again in Gobineau's view). That, combined with the linguistic relation (German really is an Indo-European language, what is usually referred to in German as Indogermanisch) between the ancient Aryans and modern Europeans, appears to have been enough.

From Gobineau, this concept of an Aryan master race is usually said to have been transmitted by the British-born German philosopher Houston Stewart Chamberlain, who also happened to be the son-in-law of noted anti-semite Richard Wagner (yes the composer). Chamberlain's 1899 work Die Grundlagen des neunzehnten Jahrhunderts (The Foundations of the Nineteenth Century) is one of the seminal works of the voelkisch movement in Germany that was so hugely influential in the late 19th and early 20th centuries and which had such an important place in Nazi, and especially SS, ideology.

For what it's worth, the Nazis did use the terms Nordic and Germanic in their ideology. The SS, especially, viewed the Aryan race as having several subgroups, of which the Nordics were seen as being the most pure (way up there in Scandinavia where its cold and mongrels don't bother you - I'm serious, this was the basic idea). Other groups included the Alpine race and the Swabian race. I'm certain there were other subgroups (about a half dozen, iirc), but I can't recall them off the top of my head.

Germanic, on the other hand, was more of a super-national category that encompassed all "Aryan" peoples, so Germans proper, the Danes, Dutch, Flemish, Walloons (after about '42), Norwegians, Swedes, and some lucky folks in Eastern Europe if they had the right qualities about them.

Most of this is taken from George Mosse's The Crisis of German Ideology: Intellectual Origins of the Third Reich. It's an older work, but it covers this material pretty well.

I can't say whether this is why the Nazis chose it, but I think Aryan is a concept with far more history and potency than any of the alternatives.

The word Arya appears in the Rig Veda, an ancient Sanskrit text, and is the root of the name of Iran. Very early on, historical linguists inferred that this meant that the ancestors of the Iranians and the Vedic cultures of northern India referred to themselves as the Aryans. They then (wrongly) associated it with Éire (Ireland in Irish), giving rise to the idea that the progenitors of the entire Indo-European language tree also called themselves Aryans. Already then it has the cachet of being an authentic symbol straight out of the distant past, kind of like the swastika, which terms like Nordic or Germanic don't have. Even I sometimes think it's sad that the word Aryan has been so thoroughly polluted, because it is genuinely rare for us to have an idea of what a people who lived so long ago called themselves.

But more importantly, by the mid-19th century, Aryan became the standard way of referring to the whole Indo-European phenomenon. And by the mid-19th century the Indo-European phenomenon was taking on increasing significance. This is a time when romantic nationalists―particularly influential in Germany in the form of völkisch movement―were looking for the roots of ethnicity in the deep past through comparative philology and folklore studies. For them the Aryans were the deepest of the deep, the ancestors of half of the world. As a result, reconstructing the Aryan language and Aryan mythology stopped being an obscure orientalist pursuit and become the hottest topic in European linguistics, folklore studies and archaeology.

This was also a time when people needed new narratives about human prehistory. Now that people understood the antiquity of the earth and of the human species (Neanderthals were discovered in 1856, and three years later Darwin published On the Origins of Species), the old, biblical accounts of Noah and the Hamites and the Japhetites and what have you weren't going to cut it. The newly burgeoning fields of racial classification and social evolutionism offered a partial framework, but they didn't do much to illuminate national histories. Historical linguistics filled the gap. By equating language families with races, you could tell very neat stories about the origins of the world's major ethnic groups, combining the familiar genealogical storytelling of the old biblical scheme with cutting-edge racist ethnonationalism. Towards the end of the century, archaeology was also pressed into service by finding material "cultures" that could be matched up to a race and a language. And again, at the very base of all this were the Aryans: the ancestors of nearly everyone from Ireland to India.

The sheer size of the distribution of Indo-European languages started to push towards the Aryans towards mythical status. The only explanations people had for the spread of languages at the time was migration and conquest. If the Aryans had managed to subdue such a large area, they must have been something, right? For a long time people had assumed that the Aryan homeland was somewhere near India, because that's where the oldest examples of Indo-European languages were. But by the latter part of the 19th century the idea that Asians had conquered half the world (and Europe!) seemed absurd. So people started looking for it elsewhere. Europe, obviously – but where exactly? Racial theorists divided Europe into a number of sub-races and bickered about which one was the original homeland, and thus the purest descendants of, the Aryans. Unsurprisingly, most Germans favoured the hypothesis that they originated in Northern Europe with the "Nordic race". By Hitler's time, this had led to the conflation of the term Aryan with the Nordic race, so that it primarily meant the ancestors of the Germans, with the vague understanding that they had, at some point, gone out an conquered most of the rest of the world.

With this remarkable story of a single race spreading itself across half the globe firmly established, it's not surprising that the Aryans accumulated fantastical attributes. The Aryans were the smartest, the strongest and the boldest of the races. The Aryans built all the world's great civilizations. The Aryans are the master race. The ruling classes of non-Aryan civilizations were Aryans. The Aryans invented all technology. The Aryans came from Atlantis. They basically became a coatrack for whatever ideas someone had about what made Europeans so great and a way of projecting their pre-eminence right back to the beginning of human history.

So I think by the 1930s the word Aryan was imbued with far, far more cultural meaning than the proliferation of scientistic racial terminology like "Nordic", "Caucasian", "Xanthochroic" etc. It will have been a word almost everybody would have heard of, and one which didn't just denote just another racial group, but evoked an image of a heroic, superior race from the distant past – the progenitors of white European civilization. And in Germany in particular it was intimately linked with the völkisch movement and the origins of the German people.

I stayed up far too late writing this so apologies if it's lacking in detail. Most of it is based on Aryan Idols by Stefan Arvidsson, which is a great book that my drowsy brain hopefully hasn't misrepresented too horrendously.

I have a follow up question, if it is too much off course I'll submit my own post.

Who exactly did Hitler count among the Aryan race? I know it was blond hair and blue eyes, but what if someone from eastern Europe had that? Hitler despised them right? So would they still be considered an Aryan?