The theme of The Mysterious Island

A notable theme of this story is the perennial one of man vs. nature. Although the stranded colonists adapt remarkably well to life on the island, their natural environment still retains an air of mystery which they can't quite penetrate. As presented here, nature isn't an object to be manipulated and controlled; it's a subject in its own right, to be respected and venerated. The men recognize this, adopting a generally humble attitude to their new surroundings. It is this humility, combined with their considerable scientific and technical skills, which allows them to thrive in what might otherwise be a dangerous environment. I would argue that this presents us with an ideal picture of the relationship between man and nature, where it's possible to use science and technology to live in accordance with nature rather than using it simply to conquer and control the natural world.

Readers are likely to pull several themes from The Mysterious Island by Jules Verne. I think one of those themes is the theme of perseverance. In an overly simplistic book summary, the book is about of group of people that are shipwrecked on an uncharted island. Out of necessity, they keep a never give up attitude and persevere through inclement weather, injury, pirate attacks, etc. The colonists simply refuse to give up and good things happen as a result.  
A second theme is a theme on teamwork. The group of colonists are mainly led by Harding, and each of them understands that each person has a role to play. They each have to work hard for the benefit of the entire group. This story really shows readers what can be accomplished when a group of people are able to work together as a cohesive team unit. 
A third theme is a theme of forgiveness. This is best illustrated with the character of Ayrton. He's a mutineer that has been stranded on an island. Based on his past life, he's a bad guy; however, he confesses all of this to the colonists and repents from his past life. Instead of continuing to punish Ayrton, the colonists accept his repentant spirit and make him one of their own.