If your LaRue/La Rue/la Rue ancestors came to the USA via Canada/New France, then it is most likely they were not from the noble classes and were fishermen, farmers, and/or trades people in France and in Canada/New France. Most settlers of New France were just "regular folks" who saw an opportunity and signed up to have their passage paid to New France where they hoped to make their fortunes. Many Americans with French surnames had ancestors who entered the USA or what is now the USA via Canada. I am one of them.
In fact, among the non-noble classes in France and New France, surnames were used rather loosely and often changed or nicknames became surnames over time. The use of "dit" names for surnames was very common in France at one time and in New France. The French word "dit" means "said/called/referred to" as in "his name is John and we call ("dit") him Jack."
For example:
1680, Quebec City, New France: Jacques de Rochelle, who was called "de Rochelle" because that was the name of the city in France he came from, was given the surname "Martin" by his neighbors because Martin was his father's first name. He then was known as Jacques Martin.
1720, New France: The son of Jacques Martin, Louis Martin, moved to live near a hill and his neighbors called him Louis Martin dit Colline. "Colline" is the French word for "hill."
1760, Montreal, New France: The son of Louis dit Colline, Robert Colline (note the "dit" was dropped) moved to the city of Montreal, where there were streets and his family and former neighbors called him Robert la Rue. Eventually, Robert's neighbors called him la Rue, also.
Those are examples of "dit" names and how surnames could change from generation to generation (making researching ones French Canadian family tree a challenge, by the way.)
So, if your LaRue/La Rue/la Rue ancestors were from New France/Canada after emigrating from France, their surname may not have been anything close to "LaRue" before they left France.
Also, in Medieval France, when surnames were just starting to be used, there was the name "la rous" meaning "the red head" or a person with a flushed/red face (in Old French) and pronounced like "la rue." Today it is spelled "la roux" and still pronounced like "la rue."
You can read a bit more about French-Canadian surnames, including "dit" names here, if you wish:
http://www.afgs.org/ditnames/index1.html
Keep in mind that most of the residents of the USA and other locations where there was large scale immigration from other countries descend from normal people and not anyone who was even close to being related to a noble in the "home country." Some also "had" to leave the "home country" or were transported because of their criminal activity in the "home country."
Of course, many of those regular, normal people, even those that "had" to leave and those who were transported, worked hard and created good homes for their families in their new homeland. Their lives and work and families are a lot more important that some noble ancestry, I think.
Best wishes