Exception is already serializable, but you need to override the GetObjectData method to store your variables and provide a constructor which can be called when re-hydrating your object.

So your example becomes:

[Serializable]
public class MyException : Exception
{
    private readonly string resourceName;
    private readonly IList<string> validationErrors;

    public MyException(string resourceName, IList<string> validationErrors)
    {
        this.resourceName = resourceName;
        this.validationErrors = validationErrors;
    }

    public string ResourceName
    {
        get { return this.resourceName; }
    }

    public IList<string> ValidationErrors
    {
        get { return this.validationErrors; }
    }

    [SecurityPermissionAttribute(SecurityAction.Demand, SerializationFormatter=true)]
    protected MyException(SerializationInfo info, StreamingContext context) : base (info, context)
    {
        this.resourceName = info.GetString("MyException.ResourceName");
        this.validationErrors = info.GetValue("MyException.ValidationErrors", typeof(IList<string>));
    }

    [SecurityPermissionAttribute(SecurityAction.Demand, SerializationFormatter=true)]
    public override void GetObjectData(SerializationInfo info, StreamingContext context)
    {
        base.GetObjectData(info, context);

        info.AddValue("MyException.ResourceName", this.ResourceName);

        // Note: if "List<T>" isn‘t serializable you may need to work out another
        //       method of adding your list, this is just for show...
        info.AddValue("MyException.ValidationErrors", this.ValidationErrors, typeof(IList<string>));
    }

}