Tinkering ever so slightly with the odds, meh. The fact that the NHL applies the lottery only to the first pick (or first three as of 2016) suggests that they still don't really get it, though. At least 2016 is a move in the right direction, but they're still leaving in place a very powerful perverse incentive. It is still way, way better to do very poorly than to narrowly miss the playoffs.
It's sad to me that there is a tipping point at which many fans would, quite rationally, prefer to lose games rather than win them. This doesn't happen in other sports except maybe basketball (but the lottery is fairer in the NBA). I don't think players truly attempt to tank, but it does make the fanship experience a bit silly when it's so much worse to be middle-of-the-pack than terrible.
Randomize all non-playoff teams' chances at every one of the first 14 picks, or weight it slightly (not as much as now). Every non-playoff team still gets a decent draft pick at least. Try to win all games with nothing to lose. Done.