On a fresh install of 8.1 update 3 of Sitecore, when I am using the experience editor on a page I created, I click on some text, and a popup menu appears near the text. On it, there is a menu item that says 'more', and when I click that, I see a button that says 'edit the related item in the content editor'.
and when I click on 'more':
You can see it says 'edit related item'
On an upgraded 8.1 update 3 (I upgraded from SC7) the experience editor does not have a 'more' button, the 'edit the related item' button is disabled (I can click it, but it does nothing).
If I swap out the core database from the upgraded server to the fresh one (but keep master/web) then it works as normal, so clearly there is something missing in my core that wasn't upgraded correctly.
I've tried copying templates, layouts, application, everything I could think of (even at the root level) from core, and using the 'overwrite' option, but I can't get it.
The best option for you in my opinion is to use Razl and compare fresh core db with your upgraded one.
Recently I upgraded project from Sitecore 7.2 to 8.1upd3 and after comparing core dbs with Razl I spotted a lot of differences, especially a lot of missing or different items under /sitecore/system/Dictionary.
The 'Edit the related item' button is here: /sitecore/content/Applications/WebEdit/Common Field Buttons/Edit related item