The "loading screen" patent expired in 2015[0]. I cannot think of any games which have incorporated this feature. Then again, on a PC, loading screens are typically so short I cannot even read the hint text they attempt to display.
The Sims 3 has an item-finding minigame in its loading screens, but (for me at least) it's pretty buggy when it comes to actually recognizing inputs (frequently freezes, and there's very little input feedback so you have no idea if it's just frozen or you clicked the wrong thing).
Similarly you could just stuff around on Bayonetta during load times. Useful on 360/Ps3/Wii U but once it came out of PC, SSD speed made that practically pointless.
Some of the Rayman games will allow you to control a silhouette of the character while loading. You can just run around a bit and attack, nothing major, but it's something to do.
[0] https://www.eff.org/deeplinks/2015/12/loading-screen-game-pa...