The name of your file needs to match the module name in the Handler configuration. In this case, your Handler should be set to helloworld.handler, where helloworld is the file that would be require()'d and handler is the exported function. Then it should work with the same zip file.

Make sure your index.js is in the root of the zipfile and not in a subdirectory.

In my case I had the name of the module matching the name of the file and the exported handler, the real problem was macOS and the zip program which basically creates a folder inside the zip file so when uncompressed in AWS Lambda engine the index.js ends in a subdirectory.

Using Finder

Don't right click and compress the directory, instead select the files individual files like index.js, package.json and the node_modules directory and right-click to compress, you may end up with a file Archive.zip in the same directory. The name of the zip file is not going to be fancy but at least it will work when you submit it to AWS Lambda.

Using the command line

You could make the same mistake using the command line with zip -r function.zip function which basically creates a zip file with a directory called function in it, instead do:

$ zip function.zip index.js package.json node_modules

adding: index.js (deflated 47%)
adding: package.json (deflated 36%)
adding: node_modules/ (stored 0%)

How to know verify your zip file

Using finder, if you double click the zip file and it uncompresses in a subdirectory then Lambda won't be able to see the file as index.js lives in that subdirectory.

Using the command line and zipinfo:

$ zipinfo function.zip | grep index.js | more
-rw-r--rw-  2.1 unx     1428 bX defN 27-Jul-16 12:21 function/index.js

Notice how index.js ended up inside the subdirectory function, you screwed up.

$ zipinfo function.zip | grep index.js | more
-rw-r--rw-  3.0 unx     1428 tx defN 27-Jul-16 12:21 index.js

Notice that index.js is not inside a subfolder, this zip file will work in AWS Lambda.

Leveraging npm commands to zip the function

So I added a script to my package to zip the project files for me just by running npm run zip

{
  "name": "function",
  "version": "1.0.0",
  "description": "",
  "main": "index.js",
  "scripts": {
    "zip": "zip function.zip package.json *.js node_modules"
  },
  "dependencies": {
    "aws-sdk": "^2.4.10"
  }
}

$ npm run zip

> function@1.0.0 zip 
> zip function.zip package.json *.js node_modules

  adding: package.json (deflated 41%)
  adding: index.js (deflated 47%)
  adding: local.js (deflated 42%)
  adding: node_modules/ (stored 0%)

Here is an advance way with AWS CLI. It will save your time in long term use.

First of all you should install and configure AWS CLI:

http://docs.aws.amazon.com/cli/latest/userguide/installing.html

1) Create an archive

$ zip -r lambda *

It will create for us lambda.zip file with all folders and files in the our current location.

2) Get role ARN

$ aws iam list-roles | grep "your_role"

It will return to us ARN that we will use with our lambda. You should create it by your hands

Example for list-roles

3) Create our lambda

$ aws lambda create-function --function-name "your_lambda_name" --zip-file fileb://lambda.zip --handler index.handler --runtime nodejs6.10 --timeout 15 --role COPY_HERE_YOUR_ARN_FROM_THE_STEP_2

We are done!