"Please find a 'personal item' hidden in the center of the island."

"I don't know exactly what it is. Bring it to me, and I will gladly hand over the island to you."

Zipline

No need to backtrack on foot. Build a new zipline, or ride what's already there.

Scavenge and sell

This surprisingly eco-friendly machine turns trash into money.

You need fuel

Keep your lantern on so you don't get lost. Otherwise, you might lose all your precious trash.

A slight 'adjustment'

Don't sell your trash at face value. Multiply its worth!

Finally...

Just follow the light. 

Retrieve the 'personal item.'