Answer – I’m not a Wiccan witch so I had to check my answer on this with a couple of Wiccans to see if I’d missed anything key!
The Wiccan name is not so much something you find as much as it is something you choose or that is given to you by a more senior Wiccan in your coven/group. The name shouldn’t just be picked at random though, it should have a meaning that links to you and be representative of who you are or who you are working toward becoming.
It seems quite common these days to pick names with strong positive associations but many Wiccans feel that relevance is sacrificed for glamour. There are a lot of eagles and wolves flapping around which suggests some witches miss out on the opportunity to think long and hard about a name which has a really personal meaning for them.
The idea of the name is that it is a personal name known only to the individual and a selected few (sometimes kept private just between the individual and the Gods) and usually kept for ritual work and magic.
I’ll just reiterate – this is a Wiccan thing and not a Witch thing. I have met very few non Wiccan witches who adopt a second name specifically for magical purposes. I don’t myself and if I’m really honest I don’t really see the point in it. However it is important to respect the motives and beliefs of others and for those wishing to embrace a Wiccan path the second name is often seen as an important milestone.