I came across this unusual surname doing family research in Kentucky. I searched other message boards and found only one older thread about the name, but the information provided is incomplete. There is no other information on Ancestry.com about the name.
The name is definitely of German origin and the modern German spelling is FEDERKIEL. It is literally feder (German for feather) + kiel (German for keel, as in the center line of a boat). Put together it means feather center or "Quill". The name could have an additional occupational meaning as someone who writes or transcribes, since quills were commonly used as writing pens from the 6th to the 19th centuries.
In my particular line, the name was incorrectly transcribed as Catherine Featherule. It took me a while to figure out the error, but once I did, I found additional records supporting the Featherkile name spelling. In my line, Catherine Featherkile married Elijah Gifford in Kentucky on 12 Sept, 1811.
Variations of the name include Featherkyle, Fetterkile, Fedderkile and Federkile.