I had this problem and found an older, archived thread without a good answer. I figured out the answer and wanted to post it here in case others might need it:

The issue is that simply accepting an invite in Thunderbird will put the event in a LOCAL calendar, which doesn't sync through Lightning with Google Calendar. The solution is to Right-Click on the Invitation in the list of emails and choose: Convert To --> Event. The event that pops up will allow you to choose the Calendar (including Google Calendar(s) if you use Lightning) that you want. You will still have to accept the event in the original email if you desire, which will create a duplicate in the LOCAL calendar. This is the best workaround I have found for this issue.