The only word that I can even think of that starts with e, ends with i, and means anything appropriate is "ennui." Of course, that has five letters, which is a big problem unless you mistyped the number of letters, which is what I'm hoping.
The word ennui means something along the lines of dissatisfaction that comes from being bored. It means that you don't have enough to do and that you find life unsatisfying. I think you could argue that the boys' ennui creates at least some of the problems since they are stuck on the island without much to do and are surely somewhat dissatisfied with the way things are...
Are you sure it isn't 'evil' with a typo on the last letter? That has always been the most obvious and highly discussed theme - the battle against good and evil. the debate seems to centre around the nature/nurture argument of psychology/religion. For example, did the evil that emanates during the boys' time on the island come from a badness that was in them since birth, or did they pick up this desire for 'evil-doing' as they grew up - from within their own associations and environment. (Family, parents,siblings,friends,school,the island itself,the other ship-wrecked boys - the beast?) Was man born inherently good - or inherently evil? All these are popular themes and ideas for discussion in the novel 'Lord of The Flies' by William Golding.
What about "envy"? The word ends in "y," of course, not "i," but it might fit pretty well as an answer. I haven't read the novel in ages, but there's struggle (right?) over who gets to hold the conch shell.
I can't for the life of me think of any five-letter words beginning with "e" and ending with "i" other than the one named and discussed by the previous poster.