I like secrets and exploration. Going off the beaten path should be rewarded. :-)

My favorite gimmick is having custom flag in Port Royale 2.
It sounds silly now, but 10 years ago... well on second thought I might still want my custom flag before start playing.
I just can't grow up!

I'm curious, which games are you thinking of where the choices matter aren't shallow? Telltale's games were initially fresh and interesting, but even back then I realised the choices weren't really meaningful.

I never thought of that, but that was nice.

Did SM's Pirates offer that?

Physics and being able to interact with objects. I spent many any hour playing Psi Ops: The Mindgate Conspiracy just mindlessly wacking enemies over the head using anything you can pick up. Sure you can just shoot them, but where's the fun in that :P

To be honest, when I wrote "shallow choices with shallow consequences", I had the Telltale games in mind.... But I'm evil :P
A few examples from the top of my head:
- Witcher 2 - depending on with whom you side with in the 1st chapter, the rest of the game is completely different. And even after that there are still quite a few choices that decide the fate of several characters
- Dragon Age: Origins - while the initial campaign is the same no matter what, you can make a lot of choices that determine the fate of characters and the world itself. The 2nd game on the other hand is the textbook example of how to fool the player into believeing that their choices matter, when they absolutely don't. Even the more important looking choices have next to 0 effect later. Maybe you'll get a letter with different words in it, but probably that's all. (Inquisiton is better in this regard, but still, many of your choices change no more no less just one ending slide =/ )
- Orwell - what data you choose to upload can have severe consequences on the life of the characters you are spying on, and on the fate of the Orwell system itself (and this is literally what the whole game is about - to chose what you want your superiors to know, and what to keep secret). I've heard that "Do Not Feed the Monkeys" is similar, but I can't comment on that since I don't own the game yet.
- Detroit Become Human - I don't have a console, but I watched everything DBH on youtube, and in this game literally everything you do (or don't do) changes the fate of characters and how the story turns out. (I've heard that Until Dawn is similar, but I don't like horror, so I didn't watch any actual gameplay, so can't comment on that :P)

Gravity Gun from HL2.

Third-person perspective:)

You can even make custom sails in SM's Pirates, but I don't think it's the game gimmick.

http://www.hookedonpirates.com/guide/flagsandsails/index.html

For those X games that do not have it, there are mods that add cockpit views ;-)

Nope. I want to play a game by myself. Idle games do play themselves!

Time loops, with the ability to carry abilities and knowledge with you.

Trains. If I can lay tracks, design rail networks and run trains all over the place, it makes my nipples explode with delight.

A non-lethal playthrough option. Dunno, guess many consider it boring, but I like stealth games anyway, so usually if a game can be completed without killing anyone (except for a few story chars maybe) then I usually go for it, just as an additional challenge. Also means a second playthrough could be entirely different which is pretty nice as well.

Games theat dynamically change each playthough (Like Prey Mooncrash) and a few older rpg's where one playthrough is entirely different from the other games with futuristic themes like flying cars and stuff.