Armenia is known for this title because (a) it's well established that Gregory the Illuminator converted Tiridates III, an event conventually dated to 301 AD, and at the same time Tiridates declared Armenia a Christian nation; (b) Armenia as a nation had a longer and more well-known 'independent' existence.

The claims for the Kingdom of Edessa or Osroene are more contested. Firstly, its existence as an independent entity only appears between 132 BC and 244 AD, and during this time moved from being under semi-autonomous status with the Parthians, to being a Roman vassal state from 114 until it was absorbed as a Roman province in 214.

There are two stories about Osroene that connect it with an early arrival of Christianity. One is that Abgar Ukkama in the first century wrote to Jesus and invited him to visit Edessa and heal him.

More credible is the story that Abgar the Great, VIII (ca. 177-212) was converted converted by Bardaisan. Bardaisan appears to be a kind of Syrian Gnostic Christian, although his 'gnosticism' was of a very different stripe to, say, Valentinian gnosticism. Problematic is that it appears to be Bardaisan himself who records this conversion, and the evidence that Osroene became 'Christian', or even that Abgar did, is held in doubt.

I just remembered the name, for those interested: http://en.m.wikipedia.org/wiki/Edessa

I would still appreciate professional comment and point of views as for why armenia is know for this title and not kingdom of edessa.