You probably want to rename the default network on the iPhone but in the past to do this you need to jailbreak the device but now has a much simpler way.
The steps are as follows:
First, you need to install the Filza File Manager application.
Then open the Filza File Manager application installed and access the path:
Then find the line StatusBarImages, if not found with the path above you can search under the path:
Next you look for the StatusBarCarrierName line and edit the network name to the name you want.
Complete the steps above you made Respring (restart) iPhone and enjoy the results.