I need to implement C# deep copy constructors with inheritance. What patterns are there to choose from?

The typical approach is to use "copy constructor" pattern a la C++:

 class Base : ICloneable
 { 
     int x;

     protected Base(Base other)
     {
         x = other.x;
     }

     public virtual object Clone()
     {
         return new Base(this);
     }
 }

 class Derived : Base
 { 
     int y;

     protected Derived(Derived other)
          : Base(other)
     {
         y = other.y;
     }

     public override object Clone()
     {
         return new Derived(this);
     }
 }

The other approach is to use Object.MemberwiseClone in the implementation of Clone - this will ensure that result is always of the correct type, and will allow overrides to extend:

 class Base : ICloneable
 { 
     List<int> xs;

     public virtual object Clone()
     {
         Base result = this.MemberwiseClone();

         // xs points to same List object here, but we want
         // a new List object with copy of data
         result.xs = new List<int>(xs);

         return result;
     }
 }

 class Derived : Base
 { 
     List<int> ys;

     public override object Clone()
     {
         // Cast is legal, because MemberwiseClone() will use the
         // actual type of the object to instantiate the copy.
         Derived result = (Derived)base.Clone();

         // ys points to same List object here, but we want
         // a new List object with copy of data
         result.ys = new List<int>(ys);

         return result;
     }
 }

Both approaches require that all classes in the hierarchy follow the pattern. Which one to use is a matter of preference.

If you just have any random class implementing ICloneable with no guarantees on implementation (aside from following the documented semantics of ICloneable), there's no way to extend it.

try the serialization trick:

public object Clone(object toClone)
{
    BinaryFormatter bf = new BinaryFormatter();
    MemoryStream ms= new MemoryStream();
    bf.Serialize(ms, toClone);
    ms.Flush();
    ms.Position = 0;
    return bf.Deserialize(ms);
}

I don't think you are implementing ICloneable correctly here; It requires a Clone() method with no parameters. What I would recommend is something like:

public class ParentObj : ICloneable
{
    public virtual Object Clone()
    {
        var obj = new ParentObj();

        CopyObject(this, obj);
    }

    protected virtual CopyObject(ParentObj source, ParentObj dest)
    {
        dest.myA = source.myA;
    }
}

public class ChildObj : ParentObj
{
    public override Object Clone()
    {
        var obj = new ChildObj();
        CopyObject(this, obj);
    }

    public override CopyObject(ChildObj source, ParentObj dest)
    {
        base.CopyObject(source, dest)
        dest.myB = source.myB;
    }
}

Note that CopyObject() is basically Object.MemberwiseClone(), presumeably you would be doing more than just copying values, you would also be cloning any members that are classes.

Try to use the following [use the keyword "new"]

public class Parent
{
  private int _X;
  public int X{ set{_X=value;} get{return _X;}}
  public Parent copy()
  {
     return new Parent{X=this.X};
  }
}
public class Child:Parent
{
  private int _Y;
  public int Y{ set{_Y=value;} get{return _Y;}}
  public new Child copy()
  {
     return new Child{X=this.X,Y=this.Y};
  }
}

You should use the MemberwiseClone method instead:

public class ParentObj : ICloneable
{
    protected int myA;
    public virtual Object Clone()
    {
        ParentObj newObj = this.MemberwiseClone() as ParentObj;
        newObj.myA = this.MyA; // not required, as value type (int) is automatically already duplicated.
        return newObj;
    }
}

public class ChildObj : ParentObj
{
    protected int myB;
    public override Object Clone()
        {
             ChildObj newObj = base.Clone() as ChildObj;
             newObj.myB = this.MyB; // not required, as value type (int) is automatically already duplicated

             return newObj;
        }
}