Where do I find the object names of icons in the FontAwesome free packages?

There are only 4 packages in Font-Awesome:

Name    | Free | Paid | Prefix | NPM Package (free)                  | NPM package (paid)
---------------------------------------------------------------------------------------------------------
Solid   | Yes  | Yes  | fas    | @fortawesome/free-solid-svg-icons   | @fortawesome/pro-solid-svg-icons
Regular | Yes  | Yes  | far    | @fortawesome/free-regular-svg-icons | @fortawesome/pro-regular-svg-icons
Light   | No   | Yes  | fal    |                                     | @fortawesome/pro-light-svg-icons
Brands  | Yes  | No   | fab    | @fortawesome/free-brands-svg-icons  | 

On the Search icons page you can filter them by package so you know what icons belongs to what package; As an example the following link gives icons filtered by Solid package.

That being said, from the code you posted you can deduct the React name of the icon adding the prefix "fa" from the icon list; the icon "Coffee" in React is "faCoffee".

And from the filtered link you should be able to find what icons belongs to what package.

I prefer to go down the route of digging into the code in Github. Namely this file https://github.com/FortAwesome/Font-Awesome/tree/master/js-packages/%40fortawesome/free-regular-svg-icons

This includes the filename, eg. faAddressBook.d.ts as a TS declaration, but gives you a good pointer to what is there, and what is not.