As Kishork noted, this particular issue was related to a custom audience filter. Just wanted to add that we've seen this same issue several times in our Sharepoint 2010 environment after using Metalogix to move content, or running various jobs. The initial symptom is the same - the target audience property isn't visible when editing a web part.
Running iisreset won't do the trick. I've seen several posts in various forums where admins have resorted to re-provisioning the User Profile Service application, which will work, but the solution in our environment has been pretty simple.
Just add a space, or any other small change, to web.config and save.
Not sure why this works, but try this if you ever run into this issue.