Go to the jungle area & collect a Tree Vine. It will be laying on the ground.
Go back home.
A cut scene will happen & your partner will tell you that s/he can make rope now. (I think.)
Go to the right corner of your cave, select Check Rope (in upper right corner).
Remove the Tree Vine from your bag by pressing down arrow.
Go to your partner, select Talk.
Select "Could you do me a favor?"
Select "I need something made."
Then select Rope.
Select how much Rope you want & in how many days you want it made. Your partner will make it for you.
When your partner has finished making the rope, s/he will tell you. You will need to go back to the Check Rope area to put the rope into your bag.