To make good on my promise, as soon as I got home I checked a few sources. They all seemed to be telling the same story, so just to verify, I dug out my copy of OED (Second Edition) where I looked up the word "blood". And voilà:
blue blood: that which flows in the veins of old and aristocratic families, a transl. of the Spanish sangre azul attributed to some of the oldest and proudest families of Castile, who claimed never to have been contaminated by Moorish, Jewish, or other foreign admixture; the expression probably originated in the blueness of the veins of people of fair complexion* as compared with those of dark skin; also, a person with blue blood; an aristocrat.
1834 M. EDGEWORTH Helen XV. (D.) One [officer]... from Spain, of high rank and birth, of the sangre azul, the blue blood.
1879 FROUDE Cæsar xi. 120 A young nobleman of the bluest blood.
It turns out that Inka was nearly right: she proposed the origin of the phrase lies in the skin of the aristocrats which was of much lighter tone in contrast to that of commoners and peasants who worked in the sun (compare "redneck"). The underlying ethnic/racial aspect of "sangre azul" is quite interesting. The only thing to do now is to find out how this phrase entered our language, especially seeing that its English counterpart is less than 200 years old. So let me just get my etymological dictionary of the Slovak language...
There is no etymological dictionary of the Slovak language. There is no comprehensive dictionary of the Slovak language, either. The last decent grammar is from 1971 (written by the great Eugen Pauliny) and the last comprehensive overview of Slovak morphology is from 1966. I guess our linguists have better things to do. Like, say, telling people what the right word for "dog leash" is (fyi, it's "vôdzka", definitely NOT "vodítko" or "vodidlo"). It's not like our many varied dialects were dying out and in dire need of recording and preserving for the posterity. No sirreeeee.
So until I get my hands on the etymological dictionary of the Czech language, let me present my hypothesis: it's the Habsburgs. Between 1516 and 1700, they ruled Spain and until 1918 they ruled most of Central Europe, including Slovakia. It is possible that the phrase "sangre azul" entered the main languages of the Austro-Hungarian empire (German "blaues Blut", Hungarian "kék vér" and perhaps Latin "sanguis caeruleus/coeruleus", too) even before it entered the English language and then continued to influence other languages of the empire, including Czech and Slovak. I think I'll check the historical dictionary of the Slovak language, too. There's - at last! - one thing the boys and girls at JÚĽŠ got right.
* fair complexion = svetlá pleť