No, it’s not possible. The exception happens because a reference is null, and references doesn’t have names. Variables and class/struct members have names, but it’s not certain that the reference is stored in either of those. The reference could for example be created like this:
someObject.GetInstance().DoSomething();
If the GetInstance method returns null, there is a null reference exception when you try to use the reference to call DoSomething. The reference is just a return value from the method, it’s not stored in a variable, so there is nothing to get a name from.
If you have debugging information in the compiled assembly, you will get the line number in the stack trace in the exception, but that is as close as you can get.