Finding the variable name passed to a function

This isn't exactly possible, the way you would want. C# 6.0 they Introduce the nameof Operator which should help improve and simplify the code. The name of operator resolves the name of the variable passed into it.

Usage for your case would look like this:

public string ExampleFunction(string variableName) {
    //Construct your log statement using c# 6.0 string interpolation
    return $"Error occurred in {variableName}";
}

string WhatIsMyName = "Hello World";
string Hello = ExampleFunction(nameof(WhatIsMyName));

A major benefit is that it is done at compile time,

The nameof expression is a constant. In all cases, nameof(...) is evaluated at compile-time to produce a string. Its argument is not evaluated at runtime, and is considered unreachable code (however it does not emit an "unreachable code" warning).

More information can be found here

Older Version Of C 3.0 and above
To Build on Nawfals answer

GetParameterName2(new { variable });

//Hack to assure compiler warning is generated specifying this method calling conventions
[Obsolete("Note you must use a single parametered AnonymousType When Calling this method")]
public static string GetParameterName<T>(T item) where T : class
{
    if (item == null)
        return string.Empty;

    return typeof(T).GetProperties()[0].Name;
}

What you want isn't possible directly but you can use Expressions in C# 3.0:

public void ExampleFunction(Expression<Func<string, string>> f) {
    Console.WriteLine((f.Body as MemberExpression).Member.Name);
}

ExampleFunction(x => WhatIsMyName);

Note that this relies on unspecified behaviour and while it does work in Microsoft’s current C# and VB compilers, and in Mono’s C# compiler, there’s no guarantee that this won’t stop working in future versions.

I know this post is really old, but since there is now a way in C#10 compiler, I thought I would share so others know.

You can now use CallerArgumentExpressionAttribute as shown

// Will throw argument exception if string IsNullOrEmpty returns true
public static void ValidateNotNullorEmpty(
  this string str,
  [CallerArgumentExpression("str")]string strName = null
)
{       
  if (string.IsNullOrEmpty(str))
  {
    throw new ArgumentException($"'{strName}' cannot be null or empty.", strName);
  }
}

Now call with:

param.ValidateNotNullorEmpty();

will throw error: "param cannot be null or empty."

instead of "str cannot be null or empty"