An obvious example is that of those who change from Neil to Nil. Similarly Lee can easily be changed to Leo or Leon. There are many other examples. Some people have second Christian names. Thus someone called Pamela Mary could simply use her second Christian name as her Orthodox name.
Some people simply have a favourite saint and have always wanted to be called by that name.
This is the simplest case of all. Others may wish to take on the name of someone in their family. Thus we know of one little Russian boy who was not baptised and did not have a Christian name. On baptism he took the name of his grandfather, who did have a Christian name. The result was that not only was the little boy baptised, but also that his grandfather started going to church, so bringing happiness to three generations.