The term is usually employed by evangelists as a synonym for conversion - and that seems to be your usage. But the passage actually says:

'no one can see the kingdom of God unless he is born again'

which seems to suggest that being born again is the start of a process of growing awareness of God and his activities in the world, rather than the act of conversion. Indeed the whole passage speaks of the action of the Spirit - blowing where He will - rather than the human response to God which is rightly the focus of evangelism. My instinct is to argue that Aionios has been 'born again' - but still needs to be converted / repent before God / accept God's forgiveness and grace - at least by his own description of himself!