The one part of King of Tokyo I don't like is when players have to leave the game because their monster dies. Are there any variants/house rules that keeps players in the game?
From a game design perspective, adding a house rule that eliminates monster deaths doesn't make much sense.
King of Tokyo is centered around two resources: victory points and health. If you take away the death of monsters, you essentially remove health as a resource from the game. This also eliminates many of the game mechanics, such as monster powers through cards, but most notably two sides of the dice have now become useless (the attack and health symbols).
Now we've eliminated one way to win the game (kill the other monsters) and 1/3 of the main game mechanic (dice rolling), plus a number of monster power cards that deal specifically with attacking and health. At the point were these changes are made, it might be time to find or design a new board game that uses mechanics you enjoy while avoiding those you don't.
As a side note, I understand where the frustration is coming from. If a player gets eliminated because their monsters dies early, having to sit around and watch this game being played while not participating is not much fun. Because of the dice rolling mechanic, this game tends more toward chaos, so you risk playing solitaire for the next hour on your corner of the table if things don't roll your way.
There are a few things you could do, but they would have a significant impact on the way the game works.
There are 2 ways to win King of Tokyo. Get 20 points, or be the last monster alive. Any suggestion that makes it no longer a game of elimination would remove the second way of winning.
The trick is that if losing all your health doesn't eliminate you from the game, you need to decide what it does mean. There are a few options of how you could handle it though:
If you have no health, you cannot earn any points or attack. Thus the only thing you can do on your turn is heal. So you are still in the game, but you are slowed down a bunch, making it harder to win.
Similar to above; if you have no health, you just skip your turn, and instead automatically heal some given amount of health. This way you aren't relying on continually re-rolling, hoping for health, to get back in. But you are still slowed down as you miss a normal turn.
When the game ends, consider your health points to be part of your score. Thus someone who has no health left will not have as many points. He could still win, but it would be harder.
When you have 0 health, damage takes away your points instead. In this way, your points act kind of like additional health. Having no health is still really bad, because then you can't win by getting enough points either.