As is already mentioned in the comments, the Gauss coordinates would nowadays be known simply as "general curvilinear coordinates" or just "coordinates".
To understand why Einstein introduces this concept at all requires understanding that for the longest time mathematics was not understood as the formal science constructed by the likes of David Hilbert at the beginning of the twentieth century. Before this Hilbertian revolution, mathematics was synonymous to solving real-world problems with mathematical methods, and to certain intuitive abstractions such as "geometrical space" or "infinitely small and infinitely large". The typical scientist (in particular, the typical physicist) would have little understanding for modern mathematical terms such as "maps" ("morphisms"), "sets" or "limits", at least in the way we understand them today.
Accordingly, the term "coordinates" for a 19th century scientist would refer to a set of examples of ways of specifying points in space: latitude, longitude and altitude, or a set of linear distances such as Cartesian coordinates (other examples based on other geometrical constructions existed but were less known). Gauss, in his General Investigations of Curved Surfaces of 1827 and 1825, started using completely abstract coordinates on curved surfaces, which are described by Einstein in the lecture you have linked. I assume this is the reason why Einstein calls them the coordinates of Gauss.
(By the way, it seems that the term Gaussian coordinates is used in another, completely unrelated way as coordinates where one of the coordinates corresponds to surfaces of constant affine parameter of a geodesic congruence. I have no idea why.)