How to get back a declined Outlook invitation

The invite should still be in the "Deleted Items" Folder. If you go there, you will see the Accept / Decline options again. Click the correct one.

Declined meetings that are part of a series don't appear to show up in the Deleted Items folder. Also, copying from someone's calendar doesn't work with private calendars. In this scenario, there doesn't seem to be a way to accept the meeting other than having someone resend it.

I went to the sender's calendar and clicked the "copy to calendar" button in the ribbon at the top (as described on this page). It prompted me to accept the invite - easy.