At least one small element of randomness exists within almost all board games' mechanics. Some are more random than others, and the balance of strategy/tactics versus randomness is in the back of every board game designer's mind. There are pluses and minuses to having more or less randomness in a board game.
As it pertains to having no randomness at all, the board game becomes nothing but a puzzle. Let's take Chess for instance, apparently professional players will all play identical moves up until approximately 1,200 moves into the game. That's because there is a single perfect solution for players to play, given that there is no randomness. There is some sort of perfect way to play, in order to win a game, when randomness is completely stricken from it.
I believe board games should always have some level of randomness, as slight as it may be. After all, that better reflects the real world. Even in a highly strategic war game, there should be some randomness. Things happen in war that catch both factions off-guard and under no fault of their own. Randomness just adds that thematic - I can't know EXACTLY how this will go - element to any game.
Now that's not to say that games can't be hurt by randomness either. There are plenty of games that I believe could be improved by negating some of their randomness, like all the board games I played growing up (Yahtzee, Sorry, Clue, Risk, etc.). For example: I love Risk and it does involve strategy: with what continents you'll go after, to who you spread your armies, to preparation for bordering threats. Though I do believe that the dice rolling for every battle may be too random; favoring something more like the tactical battle mechanics in Scythe. Obviously people's opinions on randomness versus strategy in a game differ; my father's favorite board game will always be Yahtzee, for instance.
What are your thoughts on randomness? Do you lean more in favor for complete strategy such as Chess would give you, or maybe you lean more toward the original Fireball Island? Please let me know in the comments.