Nearly every human language is named after the people who spoke it, from ancient Egyptian, Hebrew and Greek, to modern tongues such as English, German and Chinese. And then we have the language of the ancient Romans: Latin. Why do we call it that, rather than "Roman"?
The Latin language is named after the area it was spoken in — or the people that spoke it. (It is impossible to distinguish the two.) Latin, by name, is the language of Latium (Lazio in today's Italian), not the language of Rome. Alternatively, you can see it as the language of the tribe of Latins.
Latinus is the Latin adjective meaning "related to Latium". The people inhabiting Latium were called Latini and their language lingua Latina.
Why did they choose to name it after Latium instead of Rome, then? After all, lingua Romana would be a reasonable name for the language. The reason is that Rome was not such a significant city by the time the language got its name. In other words, the area of Latium and their language is older than Rome. It just so happened that a small subtribe eventually took over and the language became associated with one city.
The word latin comes from latinus, "of Latium," a region in central Italy. In this territory, around the turn of the first millennium BC, lived a tribe known as the Latins, and their language was the immediate predecessor of Old Latin.
Not coincidentally, Rome was founded in this region around the 8th century BC. Other peoples were involved as well, such as the Etruscans and the Sabellians, but the dominance of the Latin language in Rome may be explained by the perception that their neighbors were uncivilized, as explained by Clackson and Horrocks:
If the notion that the speakers of Sabellian languages were wild and uncivilized goes back as far as the eighth century, it could help explain some of the linguistic divergences between Latin and Sabellian. A Roman desire to differentiate themselves from their neighbours may have led to their choice of linguistic forms which were not found in Sabellian, and the innovation of new linguistic features. (Blackwell History of the Latin Language, chapter 2)
Rome's founding within Latium, and its dependence on the Latins, help explain why the older term (Latin) was retained to describe the language, rather than the newer (Roman). William Cleaver Wilkinson suggests the explanation that Rome considered literature "a subordinate interest" compared to conquest, law, and history.
But the more proximate cause for why we use the word "Latin" today is that this is what the Romans themselves called their language:
When we say Latin, we observe the custom of the Romans, who habitually so described their own language and literature. In both cases we acknowledge the authority of Rome. (Wilkinson, Latin Classics)
The Latin language has been founded by a nation called the Latins. Check the information below as quoted from Wikipedia:
The Latins referred originally to an Italic tribe in ancient central Italy. They were living between 1200 BC and 1000 BC. From about 1000 BC, the Latins inhabited the small region known to the Romans as Old Latium (Latium Vetus), that is, the area between the river Tiber and the promontory of Mount Circeo 100 kilometres (62 mi) SE of Rome.
On this map you can see how close the Latins lived to Rome: