There are about forty people called "Flener" in Luxembourg. My paternal ancestors researched the genealogy, and lost the trace in the 1600s. They located quite a few "Flener"s in Austria and we believe that is where the name came from (via the Austrian occupation of Luxembourg back then). A lot of Luxembourgers emigrated to the USA in the early 20th century, including at least one Flener (see below), but I ignore whether this also holds for Austria.
There are close variations of the name in Luxembourg and Austria (and maybe Germany?): Floener, Flehner, etc. None of these means anything in contemporary Luxembourgish or any other Germanic language.
I am Pierre Flener and have emigrated from Luxembourg to Sweden. My paternal grandfather was Will (Guillaume) Flener, whose father was Nicolas Flener, whose father was Wilhelm (Guillaume) Flener. One person of the tree spawned by the latter emigrated to Kansas, USA, in the early 1900s.