Can I get the name of the currently running function in JavaScript?

According to MDN

Warning: The 5th edition of ECMAScript (ES5) forbids use of arguments.callee() in strict mode. Avoid using arguments.callee() by either giving function expressions a name or use a function declaration where a function must call itself.

As noted this applies only if your script uses "strict mode". This is mainly for security reasons and sadly currently there's no alternative for this.

For non-anonymous functions

function foo()
{ 
    alert(arguments.callee.name)
}

But in case of an error handler the result would be the name of the error handler function, wouldn't it?

In ES5 and above, there is no access to that information.

In older versions of JS you can get it by using arguments.callee.

You may have to parse out the name though, as it will probably include some extra junk. Though, in some implementations you can simply get the name using arguments.callee.name.

Parsing:

function DisplayMyName() 
{
   var myName = arguments.callee.toString();
   myName = myName.substr('function '.length);
   myName = myName.substr(0, myName.indexOf('('));

   alert(myName);
}

Source: Javascript - get current function name.

All what you need is simple. Create function:

function getFuncName() {
   return getFuncName.caller.name
}

After that whenever you need, you simply use:

function foo() { 
  console.log(getFuncName())
}

foo() 
// Logs: "foo"