Why is ArcGIS called that?

In the beginning, there was INFO (a standalone relational file manager, available on minicomputers like Prime, VAX, and DG) made by Henco. Then ESRI (now Esri) added geometry processing from its initial PIOS (Polygon Information Overlay System) product , paired with attributes within INFO, and called the result ARC/INFO ("arc" files supported line types, both linear features and the edges of polygons). The name mutated to Arc/Info sometime after ArcView was released, and eventually became the legacy ArcInfo Workstation product.

The "Arc" part has been retained across time, and the current Esri product line ups the ante to "ArcGIS" in most of the core products.

There's a more complete timeline on the Esri web site.

My understanding is that in the beginning of GIS on mainframes and continuing into the minicomputer and then personal computer age, different programs were used for vector and raster processing — memory was at a premium and programs needed to remain small and focused. ArcInfo was originally designed for vectors, other programs like Idrisi were designed for rasters. Since every vector feature can be represented by a series of arcs (though effectively flat ones here), “Arc” was a simple way to signify the vector focus of the software.

According to https://en.wikipedia.org/wiki/ArcInfo , raster processing in the form of the GRID subsystem was added in Arc/Info 6 and then added to ArcView 3, which must have been sometime in the late ’90s.