This tutorial is for iDevices, not Android, I am sorry. To rename your worlds, you are going to need these:

A computer
Your iDevice
A cord that connects the two (your charger)
iExplorer (click here) or iFunbox (click here) (You only need one, and I prefer iExplorer)
A map you'd like to play

Step One:
Go on your iDevice. Open up your multitasking bar (double click home button). On IOS 7+, swipe through Minecraft PE to delete it from your multitasking bar. On a lower IOS, hold down on Minecraft PE and hit the delete symbol. Don't worry, though, this only shuts down MCPE, it doesn't actually delete it.

Step Two:
Open up iExplorer/iFunbox and go to the Minecraft PE app. Go to Documents>Games>Com.Mojang>MinecraftPEWorlds. Now choose your world from the list.

Step Three:
Drag the level.dat file from the folder somewhere onto your computer. Open it with a text editor.

Step Four:
This is what it looks like for one of my worlds:

The bold part is what you'll need. My world is currently called "My Password". Change that to whatever you want. If you make it too long though (keep it at 10 or less to be safe) it'll get cut off.

Go into MCPE and your world should be renamed!