There are several issues. When I learned about it, the main issues involve if the child was born Jewish or not. This is from memory and from knowing someone that it affected.
- If the child was born Jewish and if the biological father was Jewish, then he should be called by the biological father's name.
- If the child was born Jewish and the biological father is not Jewish then he should be called in the same way as a child not born Jewish.
- When the child is not born Jewish (even with a Jewish father), he can be called 'ben Avraham' as any ger or he can be called 'ben Ploni' (Ploni is the adoptive father and not the biological father) in order not to cause embarrassment to the family.
This last case is indeed what I have seen in a particular case. If the adoptive father is a Cohen or Levi, the child is called 'ben Ploni' and not 'ben Ploni haCohen' because the child is not a member of the shevet.
Note that a child born of a nonJewish mother and a Jewish father is not considered to have any halachic relationship to the biological father. The child is treated just like any other nonJewish infant.