Philip Marchand mentions this question in his book 'Ghost Empire'. In his view French peasants had stronger claims to their land, and newer industrial farming techniques took longer to introduce in France, so there were less unemployed farmer colonists available. (This probably applies to the 19th century more than the early 20th).
Here's an article in French thats discusses french immigration to the US.
Basically, you haven't heard of them, because they didn't immigrate in the same numbers.
Ainsi, de 1820 à 1986, 772 000 Français immigrèrent aux Etats-Unis, soit 1,5 % du total des immigrants au cours de cette période, face à 13,3 % d'Allemands, 10 % d'Italiens, 9,5 % de Britanniques et 8,8 % d'Irlandais (3)
Translation: Thus, from 1820 to 1986, 772 000 French immigrated to the United Staes, or 1.5% of the total of immigrants over this period, compared to 13.3% of Germans, 10% of Italians, 9.5% of British and 8.8% of Irish.
The author says that there was little reason to go the Americas at least not for large numbers of people. Some people left France when conditions in france were bad (French Revolution and economic recessions) but not necessary go to the US to stay and not in large numbers.
The article is pretty good although it discusses more what immigration did happen and why.
I'm French and studied the question actually. The explication is rather simple: it's our demographics. The country's transition started sooner than in most European countries and happened in such a way that we did not experience a demographic "boom" in the 19th century. Indeed, mortality and natality decreased at the same time, which is quite unusual. So, no "boom", no "over-population" and no unemployment.
It's actually a very interesting topic. Keeping in mind something as simple and as "mathematic" as demographics helps you understand many Historical events. For instance, it's a big issue when comes the time to plan a war. At the end of the 19th century, French leaders were already aware that France was falling behind demographically. As a consequence during WW1 France had to be less retrictive regarding the age of its soldiers (both younger and older). And during WW2 demography was once again blamed.
Ironically, France's demographic profil has remained quite stable over the past century, and the country is currently topping european charts while most of the rest of Europe is experiencing sub-replacement fertility rates, and Germany is even shrinking.
To piggy-back off the OP, would French immigrants have been more likely to go to Quebec than the US?
Also, remember that immigration to North America from Europe started in the 1500s, and that early on, it was mostly in the name of empire-building, not out of dire necessity as was true in the early 1900s. The French immigrants came here for the most part in the 17th and early 18th Centuries, and certainly not in the great numbers of the later migrations. These were mostly rural settlers - trappers and farmers, and so large French population centers did not rise up, except in a few places like New Orleans and other Gulf of Mexico port locations. So, families of French descent are definitely out there, but just a lot more geographically spread out than some other groups of European-Americans. In downstate, rural Illinois, and on down the Mississippi, you will find plenty of families with French surnames.
In addition to the other information, many of the French came over much earlier (before 1787) as Huguenots. They were Protestants fleeing persecution in France and went to the colonies.
Would it have helped that France unlike a lot of the main sources of colonizers had its own colonial empire making a place like Algeria a more attractive destination?