I'm working on a small grid-based puzzle game with a client and we're contemplating what features keep people enjoying or returning to games over the long term.
Mm... "grid-based" isn't painting a super great picture. That can work in a number of ways, from immensely simple 9x9 slide puzzle like things to something more abstract like, I don't know, Fez.
But
Panel de Pon (Tetris Attack / Pokemon Puzzle League / Planet Puzzle League) and
Catherine are some of the best puzzle games to ever exist. With enough ingenuity on the latter, because the core mechanisms aren't
that advanced, I'm sure a small or one-person team could adapt it in a worthwhile way. Just two randomly pulled videos from the bowels of YouTube:
The puzzle genre is so over-saturated on every level that the hardest thing is actually standing out. Lightning in a bottle aside, I don't have the foggiest where or how one would have the best shot at accomplishing that.
A good starting question here would be: Is this a question that's being asked because the client wants to make the best game possible (which ostensibly has the knock-on effect of amassing a following for future releases) or because the client is considering options for recurrent spending within a single game?
Anyway...
Unlockable special game pieces - This sounds vaguely RPG-lite progression-esque, which usually turns me off of a game. It
especially turns me off of a game if it's something that gets reset when you start over, or if your save file corrupts or something.
Personalization of guide characters (e.g. Katamari Damacy cousin dress up) - Pretty sure I've never spent more than a couple of minutes on this in 99% of games, if I've ever noticed the feature existed at all. When doing a Create-a-Character, outside of an MMO setting, typically whatever is on offer when the character is first created is what the character looks like throughout the entirety of my time with the game. I do not tend to even look at the in-game store as things are unlocked unless something in here is specifically required or there's an associated Xbox Achievement/PlayStation Trophy or something.
Weekly challenges - Kill it with fire. It's something that sounds good in theory, but immediately creates a scenario wherein I either feel like I'm being forced to deal with a time gate (Planet Puzzle League, for example, had a daily Training session - I would have preferred to do that multiple times a day when I was free, than try and remember to do it once a day)
or I suck it up and do it, but the first time I miss a day/week I immediately lose all interest in doing it any more.
Mini games - Catherine has a mini game. It's the worst part of the entire game, somehow takes the same core game concept and
makes it worse, and the time would have been better spent making another amazing stage puzzle or two to play with instead.
Unlockable additional guide characters - Mostly in the same boat as the personalization thing. Except I'll actually change the portrait/avatar/character if one unlocks that I like to look at more. ...Please for the love of Baphomet avoid the animu male gaze/Gacha nonsense. Cute? Awesome. Cool? Great. Tig ol' bitties flying across the screen? Delete!
(but it clearly works...) 
Beating a personal high score - I have to already be super into the game and its mechanics for this to matter. Alternatively, the game needs to have achievements/trophies tied to this to force my hand - but the game also has to be good, or I'm using the achievements/trophies as justification for not bothering to play it to begin with.
Beating leaderboard high scores competitively - Global leaderboards are always hacked and meaningless. Friend leaderboards can be fun...
if the game is popular and a lot of my friends play it. Big, huge If. Otherwise, don't care. I generally only compete against myself, as a means of general entertainment or to gauge whether or not I'm getting better at the game.
Environments that change with the real world season - Pissing money down the drain if you ask me. I barely noticed the seasons changing in a game like Pokemon, unless there's some sort of clever mechanic in play I can't imagine caring in a puzzle game. At best a "oh that's pretty," and then moving on. Certainly nothing that would get me to come back; if it's really good art I might Google search what the other ones look like for ~2 minutes.
Holiday themes or events that happen in-game - See above.
Features for sharing to social media - I'm not the target audience for this; I don't even
have social media. But when I did, I didn't connect any of my gaming profiles to my social media accounts (even before all the privacy and tracking stuff came to light).
Unlockable collectibles like stamps, stickers, or other non-interactive items - Extraordinarily YMMV. I care about stuff like this in something like Smash Bros., where they're accompanied by stories and trivia and the like. If the puzzle game was themed after something interesting - maybe history, or I guess since I've brought it up Catherine has a bunch of random ass trivia regarding alcoholic beverages since it's technically set inside of a bar - then this could be a nice little bonus. I don't necessarily know how far out of my way I would go to unlock them, but I would probably enjoy the ones I did and they shouldn't be too intensive to whip up.
A journal that fills up with the results of past games, that you can go back and review - I love stats. Stats are awesome. Also see above about competing against myself to see how I've improved. Basically a leaderboard for yourself, so... yeah, that's cool.
Unlockable themes or skins for the game overall - See all of the above for personalization options. If something looks nice at the start, great! Though I think focusing just on three shades - bright, neutral and dark - and making those look really good would be better.
Trivia relevant to the game's theme - ...Oh. Hah! Yeah, see above.
Platform specific achievements - Overlooking the granular stuff of what types I personally like, yeah, I enjoy achievement hunting. The biggest thing is to try and make them clever, though there is absolutely an audience that will buy a game
just because it has easy achievements. But they give zero fucks about the game itself 100% of the time, and unless you pump out lots of easy achievement games they won't remember your company.
Keeping of personal categorical stats (e.g. Total play time ever, total number of puzzles completed ever. Total number of a specific thing completed ever) - Mmm, stats. Not something to keep me coming back or keep playing, but it's amusing to look at how much I've done every now and again while playing or to just scroll through once I've finished before deleting the game.
Access to in-game tutorial documents which describe deeper strategy (e.g. using probability to your advantage, certain patterns to look for, etc.) - Always a good thing to have by default. Accessibility options too, depending on the exact nature of the game. Like don't just use colors for puzzles, match colors
and shapes (so like Red could always also be a Heart, blue could always be a Teardrop, etc) so people with various types of visual limitations can still enjoy. If audio is involved, same thing - pair the audio cue with like a non-intrusive but noticeable wave ripple effect or something somewhere on the screen.
Deep strategy puzzles (i.e. like how there are chess puzzles with a setup prompting you to make the single best move towards a goal) - Loved things like this in Panel de Pon. Basically it gives you a half-completed board, gives you a number of moves to clear the board in, and lets you work out the best approach. Simultaneously teaches strategies and trains the player to look for certain patterns during the "main" game type.
I'm sure I could wall of next more but it's late so that's it for now.
I did leave it intentionally vague. I'm curious in general too about what kinds of games first come to mind. There's already a team of people over intellectualizing what could be fun, but it's also important to hear what people do actually play, and what the scope of those games are.
