Where do the surnames Steenis and v/Van Steenis come from?

Answer

There is a very good reference work, which can be found in almost every public library, the Dictionary of Family Names in Belgium and Northern France (by Dr. Frans Debrabandere, published in a revised version by LJ Veen / Het Taalfonds in 2003). It contains, among other things, all variants and the first sources of each surname.

(Van) Steenis is clearly a geonym, referring to a place called Steenhuis. There were hamlets named Steenhuize in East Flanders, West Flanders (Avelgem) and Flemish Brabant (Vissenaken). Lesser contenders are the German hamlets of Steinhaus (in North Rhine-Westphalia and beyond, in Lower Saxony).