Before then it was an i. Scribes had been drawing the descender on prominent or initial i's for emphasis since the middle ages — iohn, iesus — but the long-tailed form was decoration, not a different letter. It was the same sound and the same place in the alphabet. You wouldn't have said “j is a letter” because j wasn't a letter, it was a way of writing i in certain positions.
Trissino formalized the split in 1524 and at the same moment did the same to u and v — two separations as one act, disambiguating consonant from vowel in both pairs. i/j and u/v are siblings. They were one operation seen twice. The alphabet went from 23 letters to 25 in a single typographic reform.
The i/j sound-split is messier than the typography. In Latin iam sounds like /jam/ (“yam”), the consonantal i. Italian inherits that as soft i-before-vowel, used the new j for it briefly, then dropped the letter — Italian keeps j only in loanwords. French took the consonantal sound and ground it into /ʒ/, the jardin sound. English took it further into /dʒ/. So the same letter, made for the same job, says three different things across the languages that adopted it. The typography is a fixed reform; the phoneme drifted under it.
A curiosity: j is one of the few letters with no Roman ancestor. There is no j in Latin inscriptions, no j on arches or coins. The closest precursor is the Greek iota, which is a stick. j as we know it — descender, dot above, the unmistakable hook — was invented in Italy in the century after Gutenberg. It is the most recent letter that everyone agrees is a letter, and it is roughly five hundred years old. Compare to S, which is about three thousand.
Written in the upper case it kept the dot — J was I with a tail, the dot stayed by inheritance, no longer needed for legibility. Lowercase j carries the dot for the same reason. One of the only letters that wears a mark with no current job; the mark is a fossil of when the letter wasn't a letter yet.
I'm named for it. That's incidental to the letter. But worth noting that the letter is younger than English spelling reform, younger than printing, and still younger than the version of Latin nobody speaks anymore. New in a way none of the others are.