Old English used gendered nouns, like French and Spanish (etc.) do today.

It also used a system of noun declensions (that is, the ending--or, sometimes, the middle vowel--changed, based on "person," e.g.: first=I, second=you; on "number," e.g.: single, dual [they had a word for "we two," and a dual "you," which functioned like "y'all"], and plural; and, finally, on "case," which is essentially how a noun is used in a sentence. So, "I see him" vs. "he sees me," where you can tell by the form of the pronoun which is the subject of the sentence and which is the object).

So, while they didn't have nouns that were as gender-specific as "sister," or in OE, "suster," for cousins, those nouns were still "marked" as feminine or masculine by their forms. That is, "níd-mǽg" meant "male cousin" or "near-kinsman." ^1 And "níd-máge" meant "female cousin" or "near-kinswoman."

But then came the Norman Conquest of 1066. William the Conquerer preferred to "import" his own nobles rather than "adopting" those in England already, so French became the language of the courtly and high-born (and rich-enough wannabes). Accordingly, lots of Old French words were adopted into everyday language (by drips and drabs at first). But many of the old words, like those for kinsman/woman, were replaced by Norman equivalents or near-equivalents.

By the 1200s, another linguistic change was taking place: early Middle English (OE, but now with more French) was beginning to lose its case system, and many gendered Old French words lost that association. By the time of Chaucer (d. 1400), both gendered nouns and the case system were mere remnants of their former selves--or had disappeared entirely. Like, for example, "cousin."

The general scholarly consensus seems to be that some words (like "brother") were just SO strongly rooted in people's experience that they were retained.

I'm away from my bookcase atm (aaahhhh!!!! Noooooo!!!!) but would be happy to give more resources soon, if anyone's interested. I do recommend playing around with THE gold-standard Old English dictionary, which has been put online: Bosworth-Toller

^1 There are actually words closer to our "cousin," but they're more confusing, so I took the easy way out. :)

/r/linguistics and /r/asklinguistics are good subs for questions like this!

To start with, cousin is a loanword from Old French and it was loaned to English during the Middle English period, around 1000 years ago. There's a simple answer here, in Old French cosin in masculine and cosine in feminine were only distinguished by a final unstressed sound which either may have been ignored when the word was borrowed or (as happened to all these unstressed sounds in English) been lost over time.

But what's more interesting is the fact that English kinship terms so greatly rely on French.

Because in fact, almost all English terms for extended family (known as collateral terms) are either Old French loans such as aunt, uncle, niece, nephew and grandparent or modeled on Old French terms, such as grandfather, grandmother, grandson, granddaughter and grandchild (grand is Old French and the second part mirrors the English equivalent of the family member in French).

What does this tell us? Basically that the entire stock of collateral kinship terms was replaced by loans. How could this happen? Well, many of these kinds of terms are related to one's place within society; your clan or even marriage.

In many cultures, maternal and paternal relatives are treated differently, you might belong to one clan or the other and you might be pushed to or forbidden from marrying family members from one side.

It just so happens that the reason why Middle English absorbed so many Old French terms in the first place was an event that came with overwhelming changes to England's political and social structures - William the Bastard Conqueror's invasion.

Kinship terms are often categorized based on patterns named after prototypical examples. Modern English has an Eskimo system, which focuses on the nuclear unit in distinguishing parents from their siblings and your siblings from their cousins, but using much vaguer terms for how these folks are all related to you.

For many languages, using the same term for your father's sister and your mother's brother's wife would be as bizarre as using the same term for your brother and your mother's brother would be for us.

Old English appears to not have had an Eskimo kinship system. The evidence varies wildly and can be interpreted in different ways. But it seems to have had what's described as a Sudanese system, with a strong focus on distinguishing between paternal and maternal relatives.

There are no clear terms for cousins, but one that appears is mōdrige, which both referred to your mother's sister and a female cousin of some sort. Another is swēor, which referred to both a father-in-law and a male cousin of some sort, probably indicating that marrying the offspring of your male cousins was common at some point (and some place) during the Old English period.

There were likely a variety of terms and cousins were probably grouped more based on whether they were of maternal/paternal relation and may have not even been considered family if they were of a certain relation.

This Sudanese system seemed to have already been weakening and collapsing into the modern Eskimo system during the Old English period and shifting towards a focus on the nuclear family.

The Norman conquest and imposition of new taxes as well as censuses (William I himself ordered the Domesday Book, a huge survey of England) likely contributed to the death of the old system and replacement with not only an Eskimo style more like what's found in Romance languages, but the usage of Romance terms in place of the old ones.

As with most English philology, a good place to begin is the Norman conquest of England. After the dramatic succession of events following the death of Edward the Confessor—events which I'm really not qualified to talk about—there was an immediate change in the spoken language of the elites and government. Since Late Antiquity, the south of Great Britain had spoken Old English, a Germanic language of close kinship with Old Norse, Frisian, Frankish, Low and High German (the ancestors of the Scandinavian languages, modern Frisian, Dutch, and the various German dialects respectively). This is the language of Beowulf, and has the reputation of being almost entirely unintelligible to Modern English speakers without close study. The new Norman elites spoke a variety of Old French, similar but not identical to the ancestor of Modern French. Over the Middle Ages, the French of the English court gradually developed local characteristics, a language variety known to scholars as Anglo-Norman; at the same time, English as spoken by the commoners gradually Frenchified, a process likely beginning in London with the court and spreading out from there. This phase of English, between the beginning of French influence but before the Great Vowel Shift comes into play, is referred to as Middle English. Its descendant Modern English has a metric crapton of French loanwords in almost all areas of life. We no longer refer to cow or pig meat, but beef and pork. Instead of rike or land, we say government and country. Candle, library, cabbage, person, chair, sausage, table, and joy are all from Anglo-Norman as well, as are (more importantly) parent, aunt, uncle, niece, nephew, and cousin.

Please permit me a brief digression. Starting with the 1871 work of Lewis Henry Morgan, anthropologists have distinguished between six different systems of kinship terminology—different ways of separating or lumping together one's uncles, aunts, and cousins. Both Latin (ancestral, of course, to French) and Old English had what is referred to as a Sudanese kinship—there are separate terms for one's father's brother (patruus, fædera), father's sister (amita, faþu), mother's brother (avunculus, eam), and mother's sister (matertera, modrige). In Latin, the offspring of each of these also received a different term (patruelis, amitinus/amitina, consobrinus/consobrina, and matruelis); the words amitinus/amitina and consobrinus/consobrina changed their ending depending on whether the referent was masculine or feminine. OE cousin terminology is poorly known, but not entirely relevant either.

By the fall of the Western Roman Empire, Latin—at least outside of the fields of law and literature—had shifted to what is referred to as Inuit kinship, lumping together parents' sisters as aunts and parents' brothers as uncles, and all their offspring as cousins. The same is true of Modern English. While Italian and Iberian Romance adopted the Greek words θεία, θεῖος to mean 'aunt, uncle', Gallo-Romance (French and company) simply began to use amita and avunculus for the same. By Anglo-Norman times, these were ante and oncle. Gallo-Romance also began to use consobrinus (masculine) and consobrina (feminine) to refer to all cousins—Anglo-Norman cosin and cosine.

Here's the anticlimactic ending to this whole shaggy dog story. Old English used to have grammatical gender like Latin and the Romance languages, e.g., wudu and wifmann 'wood, woman' were masculine but gafol, æx 'fork, axe' were feminine. In the Middle English period, this ends up being discarded. Similarly, unstressed vowels were gradually dropped—wudu eventually becomes wood. All this to say that the final -e of cosine 'female cousin' just got dropped and the word merged with cosin 'male cousin'.

Final notes:

• English gets the word parent from French, originally from Latin parens, an adjective originally meaning giving birth. This adjective had the same endings whether the referent was masculine or feminine.
• The word sibling is a native English term, but it actually passed out of usage for most of Modern English. Geneticists revived the word, which originally simply meant 'relative, kinsman' in 1903 to refer to 'brother or sister'. The word was extremely archaic at the point of its previous citation, in 1425.
• Iberian Romance gets primo/prima from the phrase consobrinus primus, i.e. 'first cousin'.
• I neglected to mention niece or nephew. These descend from the Latin neptis and nepos, the former via a Late Latin neptia. The Latin words meant both 'granddaughter, grandson' and 'niece, nephew'; they preserved this meaning into Old French (AN nece, nevou) and continue to have this meaning in Italian. Iberian Romance uses the Classical Latin term sobrinus, sobrina. Nibling is a modern coinage based off of sibling.

Sources:

• Boswell and Toller for Old English, anglonorman.net for Old French, the Middle English Compendium for Middle English.
• Professor Brian Schwimmer's introduction to the anthropology of kinship.
• M. Roy Harris, "The semantic prehistory of the Romance progeny of Latin 'auunculus-amita'", Romance Notes, 1965
• Robert Parkin, "On the definition of prescription: the problem of Germanic kinship terminologies", Social Anthropology, 1993