Okay, for the next unfortunate soul:
It is not possible to do that directly from either a game page (such as
https://boardgamearena.com/gamepanel?game=sevenwonders) or from the game list (
https://boardgamearena.com/gamelist?section=all).
However, there are two rather clumsy workarounds:
1. From a game page (such as
https://boardgamearena.com/gamepanel?game=sevenwonders) Press the big blue "Play this game now" button. You will be taken to "Play now" page, and the game will be added to favourites. The game session won't be created until you press "Create" button, so if you just wanted to add to favourites, you can stop here.
2. An alternate method is to go to the "Play now" page (
https://boardgamearena.com/lobby) and scroll down to the "FIND A GAME". Here, do not press "show me all the games" link as it will get you back to the game list, which, as we established does not allow adding to the favourites directly. Instead you have to type the game name into the search box. If you do that correctly the game will appear under the search box, and there will be they yellow heart with plus sign button, that allows adding the game to the favourites. Make sure that "Display Options" that is located in the "Favourite games" subsection above on that page does not have "Tiny icons" selected as it disables the hearts.
Rather unintuitive, but works.
Thank you everyone, who tried to help!