I noticed it's not very straightforward where the save games for Tomb Raider (2013) are stored. Depending on how you purchased the game, it's in a different location.

Note that it is not really straightforward to restore the saves afterwards (see below). For that reason, it's better to enable Steam Cloud and have your files synced to the cloud.

For the Steam version, it is located at:

<Steam folder>\userdata\<profile id>\203160\remote

Note that the ID for the game may be different, although, if you have the latest version, it should be as displayed. In any case, if you can't find it, try searching for files called profile.dat or save1.dat.

For the retail box or Origin, the save files are in your Windows user profile folder:

%LocalAppData%\Square Enix

Note that you can copy/paste the above link in File Explorer, and it will take you straight to the correct location (see here for an explanation.

Restoring the save files (Steam only)

In order to properly restore your files, first, be sure to have Steam Cloud turned off for Tomb Raider (you can turn it back on afterwards).

For some reason, you cannot simply place the files back (I don't know if this has got anything to do with Steam Cloud, but it did not work for me out of the box).

What I did was start a new game, and save it so a slot is being used. Leave the game and then replace (overwrite) the save files. When starting up again, you will now see the slot with the proper save. The "last campsite" slot will then still be empty. Load the game from the save slot, then go to the nearest camp site to restore the "last campsite" slot.