It was surprisingly difficult for me to get some sources on this. It was after his martyrdom; it's part of a kind-of standard practice within the early church, because so many names are in common, and there are no last names, so almost every important figure is given a moniker of some sorts. For bishops, the default is "of X", where X refers usually to their diocese, but sometimes to the major place of their ministry (which occasionally differs); otherwise they are given some epithet that is considered to suitably describe them.

Why is Justin the one who carries that name as opposed to several other martyrs? Clearly it was felt from early on (at least from Tertullian forward he is always known as 'the Martyr') that this was a defining characteristic of Justin's life, one that complemented his intellectual work as an apologist. I don't think you should read his epithet over and against other Martyrs, early christian epithets don't work in any kind of 'competitive' way.

As for sources, I ended up looking at the introductory notes in things like Gildersleeve's edition of the Apologies, and the first volume of Ante-Nicene Fathers.