It is possible that Udnys are related to Robert the Bruce. There is a definite Scottish connection. Here's this from an 1875 gazetteer description of a parish named Udny in Aberdeenshire, Scotland:
"The etymology of Udny is unknown. The name has been spelled Widney and Uldney, and it derives its name from the barony and family of Udny of Udny, who have possessed the estate for several generations. ..."
Also, look for the following book at www.openlibrary.org; search for Oudny, Udny, and Udnye; and you'll find the family in Udny as early as 1594.