Tatgenhorst is a fairly rare name in Germany (between 60 and 90 persons of that name). Most of the Tatgenhorsts are concentrated in the area of Diepholz, some 60 km west of Hannover.
http://www.verwandt.de/karten/absolut/tatgenhorst.html
The element HORST refers to a bush or thicket. GEN is a patronymic suffix. The origin of TAT seems to be be uncertain, it is a short form of a personal name like Tadilo; Thetger or Thaadulf. Förstemann supposes a stem DADI, maybe connected to "deed" (German "Tat"). Another possibility is THEUDA, people. Somebody seems to suggest "dad."
So the surname TATGENHORST may have referred to someone who lived near a thicket that went around Tatgen's house etc. And this was probably located in the Diepholz area.