I died, what can I do until the world ends?
Following the haunt, the game tends to become semi-cooperative, with one antagonist. In those cases, while you're not controlling a character, you could certainly offer advice and contribute to group-thinking.

I certainly think that answer, in line with previous ones, is most appropriate, but if you're desperate to house-rule an alternative, how about this "ghost" variant:

Your character remains on the board and has all their stats returned to their starting values, but cannot interact with anyone or anything (they're a ghost) and loses all their items/omens. On your turn, you can move according to your speed as normal, and then nominate a character/creature in your room and an attribute. Reduce that attribute by one on your character coaster and declare that the target character has a +1 bonus or -1 penalty in that stat until your next turn. This represents you messing with monsters by floating candlesticks at them, or the heroes fighting on in your memory, or whatever. If it's that kind of group, you could elaborate by describing in an appropriately thematic way exactly how you're causing the change. You could target a team-mate who really needs to succeed at that Sanity check, or pick the Strength of a monster you're all wailing on. Once a stat of yours hits bottom, you can no longer use it.

I doubt this variant would be over-powering. It doesn't even counter-balance the loss of a full-fledged character if you ask me. Disclaimer: I haven't tried it, but you could give it a go and report back.

In scenarios where it's every man for himself (i.e. it becomes fully competitive with no one traitor), any further interaction you have with the game would just be king-making, so you should probably concede that you've been eliminated and resort to heckling etc.