I'm very torn on the Lucky feat. It's very powerful and as a player I love it; but I also hate it both as a player and as a DM because it's so very metagamey. Whenever you have to make a decision whether or not to "use your luck", you're being pulled right out of the fantasy world and back to a bunch of people sitting around a table trying to decide whether or not to roll some extra dice. It's immersion-breaking and dissociative.
So I love it because it's very strong and versatile, but I hate it because it damages the fun of the game.
Fortunately, the problem is fixable: ask the player to declare in advance (at character creation time, although he's free to change it whenever) specifically what kind of lucky he is, and then the DM will apply the luck when needed just like any other game rule. "I'm lucky at saving throws" means Luck automatically gets invoked whenever he fails a saving throw; "I'm lucky at making precise shots with my bow" means Luck gets invoked whenever you miss on a Sharpshooter attack; "I'm lucky at anything involving redheads" means Lucky gets invoked on any miss/saving throw failure/ability check failure/initiative roll involving a redhead, even a redhead in disguise. Or he could specify a combination of these, or even "I'm Lucky at everything" to invoke his luck on any failure whatsoever (which means he'll probably run out of luck dice really fast).
This doesn't increase the power of the Luck feat at all, but it prevents it from being a dissociated mechanic, so it doesn't hurt the game. The PC doesn't have to make any decisions about luck during the game, and now neither does the player, so the player can put himself fully into the shoes of the PC and just roleplay.