Obtaining original variable name from within an extension method

You can use an Expression to achieve that, but performance-wise it may not be the best option:

public static void Log<T>(Expression<Func<T>> expr)
{
    var memberExpr = expr.Body as MemberExpression;

    if (memberExpr == null)
        return;

    var varName = memberExpr.Member.Name;
    var varData = expr.Compile()();

    // actual logging
    ...
}

Usage:

var test = "Foo";
Log(() => test);

Alternatively, if you're using C# 6.0, it can get a bit better using the nameof operator:

test.Log(nameof(test));

A better solution would be one that is leveraging the compiler abilities (specifically, the "Roslyn" compiler) and provide the member name on compile time.

Well, short answer is no. The variable names are not guaranteed to persist after compilation in unchanged form. That information would have to be somehow persisted (for example by the use of nameof()). Also, the variable name might not exist ("test".GetVarName()).

The long answer is: yes, possibly, but it's one of the most ridiculous things I've created in my life:

using System;
using System.Collections.Generic;
using System.IO;
using System.Reflection;

namespace Test1
{
    class Program
    {
        static void Main(string[] args)
        {
            var myVarName = "test";
            myVarName.Test();
            Console.ReadKey();
        }
    }

    static class Extensions
    {
        public static void Test(
            this string str,
            [System.Runtime.CompilerServices.CallerMemberName] string memberName = "",
            [System.Runtime.CompilerServices.CallerFilePath] string sourceFilePath = "",
            [System.Runtime.CompilerServices.CallerLineNumber] int sourceLineNumber = 0
        )
        {
            var relevantLine = File.ReadAllLines(sourceFilePath)[sourceLineNumber-1];
            var currMethodName = MethodInfo.GetCurrentMethod().Name;
            var callIndex = relevantLine.IndexOf(currMethodName + "()");
            var sb = new Stack<char>();

            for (var i = callIndex - 2; i >= 0; --i)
            {
                if (Char.IsLetterOrDigit(relevantLine[i]))
                {
                    sb.Push(relevantLine[i]);
                }
            }

            Console.WriteLine(new String(sb.ToArray()));
        }
    }
}