There are about fourty "Flener" households in Luxembourg, and a total of maybe 100 people with that name there. My paternal grandfather researched the genealogy, and lost the trace in the 1600s. Then we located quite a few "Flener"s in Austria, and now 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, but I ignore whether this also holds for Austria.
There are many close variations of the name, in Luxembourg and Austria (and maybe Germany?): Floener, Flenner, Flehner, etc. None of these means anything, in Luxembourgish or any other Germanic language. My father thought it could be an accidental contraction of the Austrian surname "Hoflehner", which means somebody who "rents" (in its medieval version, "lehen" in German) a farm ("Hof" in German) from the local lord.
I am Pierre Flener and have emigrated from Luxembourg to Sweden. My paternal grandfather was Willy (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.