EAGLE Names not showing up on board after footprint update
Now I found the answer myself.

When the REPOSITION tool is selected, click on the parts where the name is missing and then the name shows up. Also when a group is selected it's working, just right click and Reposition:Group.

More infos about the problem can be found here: https://forums.autodesk.com/t5/eagle-forum/changes-made-to-gt-name-in-footprint-not-propagating-from/m-p/8557215/highlight/true#M20265