In order to edit contacts in your iCloud account, turn off the iCloud synchronization in iPhone Settings and choose "keep in the phone". Contacts from iCloud will be copied into your iPhone's internal memory, so you will be able to edit them using MOBILedit Phone Manager.

Then just turn the iCloud synchronization back on and the data will sync back into your iCloud account.

When the iCloud synchronization is on, MOBILedit Phone Manager is not able to read data from the iCloud account and will retrieve contacts stored in the iPhone's memory only.

Sometimes iPhone doesn´t turn off iCloud synchronization and you have to reboot your device manually first.

The same also applies to other services like an exchange or google contacts.