What were some names or labels for those who became white people before white was invented, and why was this problematic?

In debates over the history of race and racism, scholarship in the last three decades has significantly pushed back in time the origin of the categories of “white” and “whiteness,” and related classification of human beings into “races.” Some influential works have gone back many centuries in documenting numerous medieval mentions of skin color, and its association with bodies, morality, religion, and place of origin. Both trajectories of scholarship emphasize the need to look much earlier than had been customary, at the eighteenth-century Enlightenment, and thus have shown how racial categorization and racism were precursors to scientific racism that posited biological bases of race.
While attention to whiteness had been growing, in part to counter the idea that race studies were primarily about black “problems,” one very influential work was published in 2010: Nell Irvin Painter’s The History of White People. Painter sees that characterizations of light-skinned people, in some respects, are traceable all the way back to ancient Scythians. Part of the value of her analysis is that, counter to what we usually find, the Greeks saw some people as too light, and included them in their “barbarian” category.
In answering your question about “white” as a category for people, looking at “names” and “labels” of necessity brings up issues of the symbolic association of all color terms. Part of what is problematic about what white was is precisely what it was not. In presenting names and labels for white, then, I will pair them with associated terms for black, and discuss related qualities.
At the most fundamental level, the association of “white” with “good” and “black” with “evil” is obviously problematic. In many European traditions, “white” often contained or equaled the positive qualities of “light” such as the sun, daylight, the sky, and knowledge, while “black” meant darkness, night, subterranean realms, and ignorance or obscurity; whiteness, in general, was deemed superior. Further associations are spelled out in Christian writings and visible in artistic iconography. The association of these qualities with skin and hair color increased in the Middle Ages, and was especially linked to religious differences and violent conflicts, such as the Crusades. Christians used the language of light and dark in positing their superiority over Muslims, Jews, and “heathens”—all of whom were called by a variety of terms.
Looking at medieval literature, from 500 to 1500, European writers often remarked on skin color and cultural differences, sometimes as inherited. “Christian” is most often a term for white, while “Moors” and “Saracens” (words for Muslims) are not white. “Heaven” and “hell” are also coded as “white” and “black.” In Parzival, a thirteenth-century German romance, the Black Moorish queen is emphatically not white, while her husband Gahmuret is white like “dew,” “brighter than daylight,” and the color of love. Through the union of such "black" and "white" people in a child who is baptized, the Christian baby becomes white. (Ramey 2018).
Black can change to white, and thus improve, becoming “noble” in their “blood”:

In the 14th century Cursor Mundi, when four Saracens who are ‘blac and bla als led’ (‘black and blue-black as lead’) meet King David and are given three rods blessed by Moses to kiss, they transform from black to white upon kissing the rods, thus taking on, we are told, the hue of those of noble blood. (Heng 2011: 261)

The dichotomy between Christian and non-Christian—Jew, Moor, heathen—seems most common. Associations of positive and negative qualities with groups of people is problematic because they are arbitrary. The related association of superiority, as religion is connected to body type, morals, and becomes an excuse for violence, is another level of problem for such classifications. The subsequent search for innate qualities in terms of biological traits, became even more problematic over time.
Allen, Theodore. (1994). The Invention of the White Race. London: Verso.
Bonnett, Alastair (2000). White Identities: Historical and International Perspectives. Harlow: Pearson.
Dee, James H. 2004. "Black Odysseus, White Caesar: When Did 'White People' Become 'White'?" The Classical Journal, 99 (2).
Goldenberg, David. 2009. “Racism, Color Symbolism, and Color Prejudice,” in The Origins of Racism in the West, ed. Miriam Eliav-Feldon, Benjamin H. Isaac, and Joseph Ziegler. Cambridge and New York: Cambridge University Press.
Heng, Geraldine. (2011). “The Invention of Race in the European Middle Ages I: Race Studies, Modernity, and the Middle Ages.” Literature Compass 8 (5): 258–274. doi: 10.1111/j.1741-4113.2011.00790.x
Heng, Geraldine. (2018) The Invention of Race in the European Middle Ages. Cambridge and New York: Cambridge University Press.
Lampert, Lisa. (2004). “Race, Periodicity, and the (Neo-) Middle Ages. MLQ: Modern Language Quarterly, 65 (3): 391-421.
Ramey, Lynn. 2014. Black Legacies: Race and the European Middle Ages. Gainesville: University Presses of Florida.
Ramey, Lynn. 2018. "Race and Identity in Medieval Europe." July 20. Black Perspectives. https://www.aaihs.org/race-and-identity-in-medieval-europe/
https://www.aaihs.org/race-and-identity-in-medieval-europe/

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