I don't know if this would fall under analytics or not, but something that needs vast improvement is knowing how much to pay players/free agents.
For example. I spent this evening comparing Florida's roster to our roster in pts per game divided by their cap hit in millions of dollars for each player. The only Florida player less than 0.1 p/m$gp was Hornqvist at 0.092. Right now we have Boeser at 0.098, Pettersson at 0.07, Myers at 0.043, Hamonic at 0.037, Poolman at 0.034, OEL at 0.027, and Dickinson at 0.052. Boeser and Pettersson are due more to struggles, but the others were all examples of inefficient contracts.
Obviously I would want to do this over a full season, and points aren't everything. I'd also want to do it other teams, to get a more exact grasp of what good value is for defencemen. But spending 30 minutes on Natural StatTrick, Cap Friendly, and Excel I was able to get a rough value on how much you can increase a player's cap and still get value. Carter Verhaege will still have great value when his extension kicks in next year. While Horvat is at 0.11 right now. He very well could not be worth his next contract. Imagine what professionals could do with more time.