I have a Logitech M510 Mouse and after every update I have to repair my mouse. Open the Logitech Control Center in System Preferences. Make sure it is Version 3.9.3 (top right of window). If not you can download here:
http://support.logitech.com/en_us/downloads
Click on Open Unifying Software button then Advanced.
Select the device (not the top but the indented device.
Select Un-pair
Now you should be able to pair the device.
Move the USB connector as close as possible to the mouse.
You might also have corrupt plist files. In your User's Library/Preferences, delete all the com.Logitech....plist files.
Log out.
Empty the Trash
Now try to pair the device.
com.Logitech.Control Center.Assignments.registry
com.Logitech.Control Center.Daemon.plist
com.Logitech.Control Center.Device Manager.plist
com.Logitech.Control Center.Device Settings.registry
com.Logitech.ControlCenter.DeviceManager.plist
com.Logitech.Unifying.Assistant.plist
com.Logitech.Updater.plist