None of the above helped me with solving the issue, but after a long time trying to deal with it before flashing my Rom all over again, I did the following:
(Samsung Galaxy Note 10.1, Revolution HD 7.2.1, Android 4.1.2, latest Avast and Antitheft as of June 2013, device rooted, busybox, Titanium Backup, SuperSU)
In Titanium Backup, scroll down to Google Play Services, in my case it was the latest v3.1.36, click this and in the popup click Uninstall.
It will ask for confirmation if you are sure you want to uninstall, as it is a system app. Choose Yes.
Exit Titanium Backup, and open this link with Google Play Store and reinstall Google Play Services:https://play.google.com/store/apps/details?id=com.google.android.gms
No reboot nessesary, just enter Avast settings and reinsert your account details to connect it.
I had the initial error repeat once, but switched my device to Data instead of Wifi, and retried once.
This did it, as it connected fine.
No need to clear data or cache for any Google apps or services, and the procedure will not delete or clear any personal data or account settings.
Remember that if you have any access/connection filtering/permission-blocking app on your device, dont block the above Service as it is essential for Avast and its cloud/account features to work as intended.
I hope this will help on your issue with Avast unable to connect to your account and/or Google Cloud Messaging Services.