The central mechanic or mechanics you choose become the core of your game. In a way, the core is your game.
Once you've worked out your core, remove everything else from your game, other than what's needed to support and enhance the core. I want to be playing your amazing core, not a whole lot of other stuff.
Dominion was the first "deck-builder" game (each player starts with a weak deck, and uses its cards to buy more and better cards to shuffle into it.) The designer originally had this as the core of a much larger game, but, in the end, he removed all that extra stuff, and the game was a hit.
Many designers think they need to make an epic game around their core, but the correct choice is often a relatively simple, half-an-hour game. Each irrelevant or extraneous thing you add, makes your game worse.
If your game is about one thing, and that thing is great, then your game is great. If it's about ten different activities, then they all need to be great, for the game to be great.
There's a second advantage of just focussing on the core of your game.
Imagine the following game (I just made it up):
There's a ten by ten grid. On each of your turns, you add a stone of your colour to the board. Players cannot place stones adjacent (including diagonal) to their opponent's stones. The player with the most stones on the board at the end wins.
It's nice that this game has such simple rules, and can be understood so easily. You can start strategising immediately.
The game above is an interesting puzzle, precisely because it's a very simple and clear problem. It's not a sprawling array of mechanics, but poses a very stark and simple question to you. If this example interests you, it's not going to be made more interesting if I add extra things on top of it.
I call this clarity. The more your game is about its central challenge, the more clarity it has.
This game is not interesting enough for me to want to play it twice, but it is interesting momentarily.
Give the players a distilled experience of one simple concept.