A very great variety of first names are given to children by their wizard parents, some of them being what we might think of as Muggle names (e.g. James, Harry, Ronald), others giving a distinct flavour of the personality or destiny of the bearer (e.g. Xenophilius, Remus, Alecto).
Some wizards have a family tradition of names. The Black family, for instance, like to name their offspring after stars and constellations (which many would say suits their lofty ambition and pride). Other wizarding families (like the Potters and the Weasleys) simply pick their favourite names for their children, and leave it at that.
A certain sector of magical society, however, follows the ancient wizarding practice of consulting a Naming Seer, who (usually for a hefty payment of gold) will predict the child’s future and suggest an appropriate moniker.
This practice is becoming increasingly rare. Many parents prefer to ‘let him/her find his/her own way’, and dislike (with good reason) receiving premature hints of aptitude, limitations or, at worst, catastrophe. Mothers and fathers have often fretted themselves silly on the way home from the Naming Seer, wishing that they had not heard the Seer’s predictions about their child’s personality or future.