1. Cancel the NHL draft (picks would just all move back a year, so if you have 3 1sts this year, you get them next year instead).
There was an opportunity in 2005 absolutely wasted. They could have simply cancelled the draft and changed the minimum draft age to 19*
*Every year, up to three to five players at 18 are NHL ready. Sometimes none are ready. Perhaps if you win a draft lottery, you can draft an 18 year old.
Now they can do it this year. The teams that would lose out are obviously those teams which would have beneficial seeding, like the Red Wings. But the NHL can compensate teams seeded in the top 5 with extra picks.
This would solve two major issues in the development system. The first is the shortsighted rule of being unable to graduate junior players to the AHL right after they're drafted. The CHL and the NHL would both welcome this change, for the CHL would get more competitive only losing its absolute top stars, and the NHL could move players they've drafted in the CHL to their AHL team right away.
The second major issue solved would be to those teams stuck between a 6th and 15th overall pick who might get players half as good as those players drafted in the top end of the draft. It's really no secret that drafts are extremely top heavy in the NHL, which means you could be drafting franchise players in the first two overall picks, and by the sixth pick, you're drafting players more likely to be third line role players. Changing the draft age to 19 would make it more likely to draft the draft's best talent in the first round (and perhaps make the 6th and 7th round more redundant).
When there were 21 teams in the league, the draft worked, but when there were 23+ teams, the draft no longer worked for anyone drafting outside the top 5 but missing the playoffs. Some teams would spend up to a decade outside of the playoffs due to their inability to acquire franchise players in the draft.
The idea is that guys like Brayden Point and Johnny Gaudreau wouldn't have been picked in the 3rd and 5th round, but they'd have been a lot higher on the scout lists.
2. Keeping up with the competitive integrity of evolving playoff systems around the world
Esports are starting to get it with the invention of gauntlets and loser brackets.
The league has to simultaneously act on the goals of entertaining its audience with "flipped scripts" (upsets) and giving them a great final chapter.
Unfortunately, great Stanley Cup Finals are few and far between. Every time there's a flipped script, and every time that two of the best teams in the league are in the same division, your most competitive series' from eventual champions happen in the second and third round of the playoffs. Obviously that wasn't true between the Blues and the Bruins, but let's suppose that your league two best teams are randomly picked teams: ~50% of your years, those two teams are going to be in the same conference, which means your most competitive series happens in the final four.
Now that the NHL has to come up with creative solutions - NOW is the time for innovation!
The "loser's bracket" is a system they should implement so that the competitive integrity of the Stanley Cup Finals is intact - and you don't have four or five game washes (or you'll get them less frequently). In the loser's bracket, outside of the Finals, a team has to lose a series twice to be taken out.
The "bye" system is designed to reward teams for finishing first in the regular season. It doesn't make sense that teams that finish first have to work just as hard as teams who finished 8th - that everything plainly hits reset and you theoretically play harder teams.
Two bye systems I've seen out there made sense: the first is allowing teams to skip into later rounds, but the second is (because teams that aren't playing can often go a little dull) to give them a game in hand when they do win a bye. That means they have to win 3 vs the other team needing to win 4. I'd only apply it to teams with a double bye.
The other thing I'd do for certain is when there are only four teams left: the league would be reseeded so as to not restrict conference - ie, the top team left from reg season stats would pick who they play out of the remaining teams: this would all but ensure, to the best of our abilities, that the Stanley Cup Finals is going to be the two best remaining teams.
These aren't foreign systems that I made up on the spot: they're systems being practised by various esports leagues. The only problem is that leagues with existing systems are reluctant to change those systems, perhaps comparing themselves to other leagues or worried about alienating viewers. The entertainment industry, however, isn't where you want to be traditional, since much of it survives on novelty. Bracket systems are an evolving breed and the NHL should use this year as an opportunity to experiment.