Researching the world of tabletop gaming I became entranced by the elegance with which some of them create an experience using only a limited set of mechanics. Boardgame design strives by its nature to limit the need for consulting the manual, doing math, tracking things. Only to present a sweet crunchy puzzle of tense anticipation.
ALLIANCE AGAINST THE LEADING PLAYER
You can give the game a theme, an elegant mechanic to invoke the thematically appropriate emotions: the joy of building things, the dread of depleting resources, the uncertainty of combat. But if we are talking about a strategy with more then 2 players, it is likely that most of that will be overshadowed by the last few turns when the more logical approach is to do harm to the player who is the most likely one to win. On one hand, there is something nice about having everyone team up against a common enemy (player with the most points), but I found this degrades the experience overall. Many games I actually like have this flaw: Twilight Imperium, The Rising Sun, Root.
MODIFY A SET OF RANDOM ELEMENTS
If you try to upgrade your character/army/whatever, granting it a higher yield, this may end up with too much math to do. Your town produces X Useful stuff every turn so don't forget to add it. The solution is to modify a set of elements that take effect when activated. Deckbuilding games are made around that idea, Monopoly has a similar concept: instead of adding cash every turn, you increase the chance of getting it by capturing slots.
It is always more fun when instead of taking turns the event happens in an instant. For example: "two players rolling a dice and the one with the higher number wins" is more fun than Player 1 rolls the dice, resolves his result, then player two rolls his dice.
For the player's strategy, some actions are more critical than others. It is best when rules accommodate the "I need to get this done no matter the cost" panic and have rules that allow performing otherwise suboptimal actions, sacrificing more resources to get something done.
TURNS AFFECT EVERYONE
During one player's turn, he may select an action everyone takes. So waiting for another player is not completely passive. Think Starlight Imperium.
TURNS ALLOW PLANNING
Usually, games get this right. But every now and then you encounter a fancy gameplay loop where the situation on the board dramatically changes right before the start of your turn, throwing all plans in the bin.
Can be great balancing force. Usually, it is a choice between increasing your power and getting points. So getting a victory point sets the player back a bit.