I just found my first memento, it appears that they are some sort of bonus to character skill in the form of star/face (mine game me 10 max cleanliness for Mary) and got it from regular gacha. They can be found in normal gachas which can be bought using coins from games or promotional gachas which can be purchased by gems from weekly quests or achievements (i'm not very sure about luxurious gachas).

Tips:
Open gachas with the only the girl that requires the memento.
Save money for when you need mementos (lev 7 mary and lev 6 sakaki) because if you open the gachas before the requirement then it doesn't count(yes you can get mementos even if there is no level up requirement).
If your girl is at max affection just play only games to get coins for gachas, affection mostly be recoved if you play games well (to get max money you have to fill the game bar till the end, beyond that there is no bonus, exept if you find extra coins but you get 50 coins for playing well 1 game=2 minutes and 5-10 coins/minute so it's probably better to play more games instead of fewer longer games; reaching the 3rd star means maximum hapyness and affection but not maximum money)

Similarly try to avoid completing weekly tasks early, you might level up a girl (mary at level 6) that requires you to complete a weekly task and you have to wait another 7 days(of course forgetting to take the rewards on time is worse).
On a side note weekly dates aren't really dates, these requirements just complete themselves every 7 days without any requirement from you(except probably you logging in daily?).