Error 'Your session expired' on Gmail Integration Side Panel with Salesforce

The Gmail Integration side panel session mirrors the session in the opened Salesforce.com tab. When your session times out in the Salesforce.com tab, it will also log you out of the Gmail Integration side panel.

However, if you refresh the Gmail page, the error does not show up anymore. You can also click on OK on the error message to refresh the side panel.
 

As a work around, since this is an issue where the session expired, you can try extending the session timeout for the affected User's profile to avoid the error message by following the steps below in your Salesforce instance:

Classic View

1. Go to Setup | Manage Users | Profiles.
2. Select the affected User's profile.
3. Select Edit.
4. Scroll down to Session Settings.
5. Change the Session Times Out After to a longer period of time.
6. Click on Save.

Lightning View

1. Go to Setup.
2. In the Quick Find box, enter Profiles.
3. Click on the affected User's profile.
4. Select Edit.
5. Scroll down to Session Settings.
6. Change the Session Times Out After to a longer period of time.
7. Click on Save.