I believe that the idea of "a new song" has many implications:
It is anti-ritualistic: When ritual is performed over and over again, it becomes an old song. Emphasizing that the psalm is a new one means that it is not ritual. It is pure glorification, pure praise directed towards God, not some old mechanical routine that worshipers perform for the sake of gaining credit with their fellow men.
It is anti-pharisaic: the "old leaven" of the pharisees is an old song while the "new leaven" of the covenant in Christ's blood is what the Psalm is about. Of course this is a NT-inspired interpretation but I believe that some of this meaning is already present in the OT text even if on an unformulated level. One way to put it, perhaps, would be to say that the NT-inspired anti-pharisaic take on this "new song" idea is a recasting in the zeitgeist of the Ist century AD of the originial anti-ritualistic meaning of the expression.
It points to the infinite nature of God. No finite set of songs can capture His greatness, forcing worshipers to endlessly invent new ones.
It point to the Holy Spirit inspiring the Psalmist. Only on the rock-solid basis of true Faith can the Spirit truly inspire one to make an accurate and beautiful song about God from scratch. Making a new song is a risk, like walking on water. Accepting and even embracing that risk makes it plain that one relies on God through Faith and the Spirit.
I appologize in advance to all those who will no doubt think that this answer has no place here because it is "theological" and not sufficiently grounded in technical Hermenutics. Well, that is true but I believe that it is sometimes useful to allow ourselves some freedom to roam around a bit in order to get in touch with the feelings that the Biblical text might evoke. After all, its meaning is grounded in those feelings. As an aside, although the above looks like Christian theology, I do not consider myself a Christian.
Also a historical note: I have never seen that the idea of "a new song" was a common thing to say in ancient times, quite the contrary. Ancient poets emphasized that their songs were "from of old" to emphasize their quality, even though they might have composed them recently or, more often, "improved" an earlier version. Thus, the idea of "a new song" has no place, for example, in Homeric poetry or the Norse Sagas or the Vedas.