Firebase functions cannot deploy : SyntaxError: Unexpected token function

As of September 2019:

1. Update firebase-admin : npm install --save firebase-admin
2. Update firebase-functions : npm install --save firebase-functions
3. Add "engines": { "node": "10" } to your /functions/package.json

...
"dependencies": {
    "firebase-admin": "^8.5.0",
    "firebase-functions": "^3.2.0"
  },
  "devDependencies": {
    "tslint": "~5.19.0",
    "typescript": "~3.6.2"
  },
  "engines": {
    "node": "10"
  }
...

As of August 2018:

Cloud Functions now support Node 8 (8.11.1). Check out this blog post.

Upgrade to Node 8

As suggested in this blog post, follow these steps to upgrade to Node 8:

1. Upgrade your firebase-functions version via npm install --save firebase-functions@latest
2. Upgrade firebase-tools via npm update -g firebase-tools
3. Add "engines": { "node": "8" } to your /functions/package.json

If you are still having the issue on a recent version (such as node 12), use the ecmaVersion parser option in your .eslintrc.js file.

Here's a sample:

module.exports = {
  root: true,
  env: {
    es6: true,
    node: true,
  },
  parserOptions: {
    ecmaVersion: 8,
  },
  extends: [
    "eslint:recommended",
    "google",
  ],
  rules: {
    quotes: ["error", "double"],
  },
};

h/t to Dean for the original suggestion.