Even if I was a bit tired, I decided to go anyway because it was his last night in my town (he lives and works in another city and he comes here only when he's onduring holidays).

We played several games, so I want to talk about the two games that I liked the most.

Obviously, you have to be careful because there are some players who check that no-one iwho’s cheating!.

Last night a friend of mine invited me to his house for a board game night.

No errors, just a thought:
"to play board games" or "to play some board games" is a more informal way to express this. If it was an event that was planned and announced in advance, then "a board game night" would be precise. If he just called or texted you earlier in the day and asked if you wanted to come over to a gathering that was organized spontaneously, then the more casual phrasing is appropriate.

Even ifthough I was a bit tired, I decided to go anyway because it was his last night in my town (he lives and works in another city and he comes here only when he's on holiday).

"if" is more useful for hypothetical statements. Here "Although" or "even though" work best.

We played several games, sobut I want to talk about the two games that I liked the most.

it's actually a contrast not a cause and effect relationship

The first one is called Lercio, and it's based on the homonymous Italian satirical newspaper of the same name.

There is nothing wrong with using "homonymous," but it is so rarely used that it stands out from the rest of the text.

One player (the editor-in-chief) chooses 3three newspaper headlines with some gaps to be filled in.

The laptops are then picked up and mixed by the editor-in-chief and read aloud.

"picked up" has a strong physical connotation, so the focus is on the action of lifting and moving them.
"collected" is more neutral, so the focus is on bringing them together and closer to the editor.
This is my impression, and others may disagree.

It's a very funny German game in which players need to get rid of their cards following a similar logic to the more famous Uno gamethe same way they do in the more well known game, Uno.

You might delete "more" or "the more famous." If Uno is that well known, then maybe those words aren't needed. There is nothing wrong with calling Uno "famous," but fame involves much more than being "well known."