Favorite mini-game is Chocobo Racing, by far. In all its iterations, from FFVII, FFXIII-2, and FFXV.
Favorite Side Quest...
Well, that one is hard to say. I could evaluate that by the feeling of accomplishment and joy at succeeding said side quest... If that's the metric I were to use, I would say the Pitioss Dungeon of FFXV.
...However that was also the most hellish, frustrating, insane, and bizarre side quest I've ever seen in Final Fantasy. But at the same time, it's stuck with me and is one of the most memorable in all the series. So clearly it did something right, despite it's ridiculously difficult and jarring execution. It's stuck in my head as one of the most memorable and challenging extra side quests of Final Fantasy.
In terms of just over all enjoyment and just fun in general... I would have to say it would be taking down optional super-bosses. The more traditional side quests, of Final Fantasy.
Now it's especially hard to choose which one.
It's like a 5 way tie between FFVII's take down of Ruby and Emerald Weapon for the Kalm Traveler, The Deepsea Research Facility and Ultima Weapon from FFVIII, XIII-2's Gilgamesh in the Colosseum, the Last Ones quest and Erishkagal from Lightning Returns, and Solheim's Omega from FFXV.
Those optional bosses and their attached side quests were the most fun and challenging to me.