Pass a parameter to FileReader onload event

Use Blob API

Using the new File / Blob API it can be done much easier.

Your second code block - also solving your problem - could be rewritten as:

for (let file of files){
 (new Blob([file])).text().then(x=>console.log(file.name,x));
}

The Blob API uses Promises instead of Events like the Filereader API, so much less troubles with closures. Also the Arrow function (x=>) and iterator for of, make the code much more concise.

Check support.

Use Fetch API

Also using the Fetch API Response Object it can be done more easy.

for (let file of files){
 (new Response(file)).text().then(x=>console.log(file.name,x));
}

Note of the missing Array [] in the constructor.

Also check support.

Try the following:

var reader = new FileReader();
reader.onload = (function(theFile){
    var fileName = theFile.name;
    return function(e){
        console.log(fileName);
        console.log(e.target.result);
    };
})(currFile);   
reader.readAsText(currFile);

Here, you're creating a new fileName variable each time a file is passed to the outer method. You're then creating a function which has access that variable (due to the closure) and returning it.