I have an upgraded site in SharePoint 2013, which I brought over from SharePoint 2007.  In this portal, there are pages with an existing Members web part to display current members of that particular site.  The members listed are hyperlinked to something.  Some launch to a "The webpage cannot be found" 404 page.  Others display old information (job titles, etc).  Now, I know this isn't something that broke when upgraded to 2013.  It's just been brought to my attention.  Yesterday, I set up the profile sync, and I can see the number of user profiles have climbed from 56 to 272.  I followed this technet article.
http://technet.microsoft.com/en-us/library/jj219646.aspx#proc1
Now that I have profiles, how do I fix the links in the Members web part to display current information?  Where can I see this list of user profiles to know what information was synced from AD?

I can answer the last question, in the User Profile Service > View Profiles you can see what info has been imported from AD. Then, you also need to check the UserInfo table of every sharepoint content db to see what info was sync'ed between the user profile service and SPF profile.

Thanks for the reply. In Central Admin, I can view/verify the user profiles are current. Now, how to get that information to display in my site. Funny that now this was reported, we received another call where the user says, "My name changed a year ago, and it displays my old name". Vague, but it's the top of the page where it displays the currently logged in user.

For out of date information, take a look at http://thesharepointfarm.com/2014/01/troubleshooting-user-property-issues/

For the members web part, my assumption is them site host URL has changed, thus the 404.

Hmmm.

stsadm.exe -o sync -listolddatabases 0
displays the jobs last ran last evening at 6 PM, yet users still show old information. 

If I run stsadm.exe -o sync -deleteolddatabases 0, do I immediately need to run a full sync, or what will happen with end users actively in the site?

Run the timer job <UPA Name> - User Profile to SharePoint Full Synchronization. This will update the UIL of each Site Collection in all content databases attached to the Proxy Group o

Just manually ran "User Profile Service application - User Profile to SharePoint Full Synchronization".  This task is scheduled to happen every hour.  This doesn't seem to have had an affect on the site. 

Are these users on the UIL not updating actually active? Do they map correctly to their Active Directory user account that is not in a disabled or deleted state?

Yes, they are active users. One of the people sits right next to me here. Her login on the top of her page shows her old name. I just created the search center and her login looks correct in that site collection. If I add her to the permissions to a page, when I search for her, it displays her correct name. I'm not sure if that's simply going out to AD at the time of the addition, or if it's looking to the profile service for information.

Yes, it goes out to Active Directory.

Is the UPA Proxy associated with that Web Application?

If you use Set-SPUser -SyncFromAD, does it update her information on that Site Collection?