The "ancient" root cause of why Gangnam turned out to be what it is today (most expensive real estate, a key business district, highest concentration of all the desirable services/shops/etc, trendiest restaurants) is simply because that's where some of the best high schools are located.

Before, the high school system revolved around an exam system and an entrance exam system where if you were smart, you could attend whatever school you chose to sit in for an exam and pass, anywhere in Seoul. It happens that some, but not all, elite high schools were in Gangnam, probably because it was actually cheaper to build a school there since Gangnam used to be the outskirts of a smaller Seoul. One day the government decided that they wanted to standardize all high schools, level the playing field, so all high schools became equal and the only criteria for admission is where you lived.

Hence the birth of school districts. You lived in a district, you went to one of the schools in that district. However, the 8th School District, the one in Gangnam-gu, just happened to still have the best schools and it seemed that the schools themselves, the parents and the establishment (who all came from those schools) sort of wanted to keep them that way. All the college prep institutions and the support systems needed for helping students get into top colleges were all there already anyway. So the 8th School District became the place to be if you wanted your kid to go to a top college. Self perpetuating cycle begins here. All the upwardly mobile parents moved there, jacking up prices of homes and rentals. Since all the well-to-do people or those aspiring to be well-to-do all lived there, all the shops and services started opening there too. That pushed housing prices even further up. All the luxury brands moved in. Then there was the commercial district in Gangnam that the city of Seoul had been trying to develop (to relieve the congestion and crowding of the Gangbuk - the traditional downtown) suddenly became attractive to corporations. All the hotels opened up shop here. Suddenly this place was the perfect storm of Luxury housing + Top tier schools + Best shopping + Best restaurants + Large corporations.

So you you end up with PSY's Gangnam Style, where everything cool, new, chic and trendy is of, from, by, in Gangnam.

Miscellaneous:
Gangnam in the broadest sense means south of the river
Gangnam-gu, in the narrower sense is a specific district (Gangnam-gu), which is one of the districts south of the river, but the central and most expensive one
Seocho-gu and Songpa-gu flanks Gangnam-gu in the south of the river and are getting just as ritzy.
However, some other gu's south of the river (Kwanak-gu, Kuro-gu, Youngdeungpo-gu, etc) have yet to become swanky, yet.

By the way, the Really, Really rich people still live in the North of the river, in the traditionally rich enclaves (Pyongchang-dong, Samcheong-dong, Seongbuk-dong, etc.) These are the places you find mansions (not posh condos like in Gangnam), and these folks don't really care about school districts because they send their kids abroad, or have other means to send them to good schools.