Strategy board games have been around for thousands of years, and come in a dizzying variety. With so many games already around, the best place for a designer to start is with the classics. Play strategy games you know and love and try some new ones. Find out what works and what doesn't, what rules you like and what could stand to be changed. Steal ideas from the greats, then make them your own.
Play as many different strategy games as you can. There are classic strategy games, such as chess checkers; war games, like Axis & Allies and Risk; and many others. Write down what you like about each game. Try to get some ideas.
Decide if you want a chance-based game or a pure strategy game. Most war games have an aspect of chance to them, but the great strategy classics, like chess and checkers, are played without dice. Using dice lets you add an element of unpredictability to the game. Playing without dice lets the players rely on their skills to win.
Base your strategy board game on one or more of those games. Borrow rules and concepts freely, then make them your own. You might, for example, like the combat system of one game, the turn-taking system of another, and the setting of a third.
Come up with a simple mock up. Sketch out a board with pen and paper on cardboard, or use the board from one of your other games. Use bingo game markers, coins or stones as pieces.
Recruit some friends to help you test your game. Have them play with you, then get their reactions. Discuss ideas for ways to improve the game with them. Change the rules and play again until you get it right.
Make the game look nice. Use paints, markers or pens to decorate the board. Buy miniatures for the game pieces. Laminate the board to preserve it.