Hi Neo and @ITCrowd,
Thanks for the suggestions. I have never configured dirsync myself, Windows Server 2012 Essentials did that for me. When I create a user in the dasboard on the server, the first and the last name are combined to a display name. Every hour (I believe) changes are replicated to Office 365 and only in that direction.
The AD user object has nothing to do with it. By default the display name attribute is even empty, filling it out does not change anything.