If you have a current iCloud ID that is a newer version of the old ID (and not an entirely new ID), go to https://appleid.apple.com, click Manage my Apple ID and sign in with your current iCloud ID. Click edit next to the primary email account, change it back to your old email address and save the change. Then edit the name of the account to change it back to your old email address. You can now use your current password to log out of Find My Friends. Then go back to https://appleid.apple.com and change your primary email address and iCloud ID name back to the way it was.
If the old ID is not an older version of your current iCloud ID, you'll have to contact Apple for assistance resetting the password, either by going to https://expresslane.apple.com, then click More Products and Services>Apple ID>Other Apple ID Topics>Lost or forgotten Apple ID password, or by contacting the Apple account security team: http://support.apple.com/kb/HT5699.