The Bettencourt name in Madeira does come from the Bettencourt family of France.Spelled many different way as some of you show.
In Portugal, like some other countries, a person is given a first name and later will pick a last name usually of a family member.They may even change it as time goes on, so it can get confusing to say the least.It is not unusual for a husband and wife to have different last names.They could even have several children all with different last names.
There was a big migration of people from Madeira to Hawaii in the late 1800's. From Hawaii they came to the U.S.
There is a great web site run by the Madeira Archives, http://www.arquivo-madeira.org/homepage.phphttp://www.arquivo-madeira.org/homepage.php On the top right click on English Version.You can look up Marriage Records as well as passport records from this site.Marriage records go all the way back to the 1500's and up to 1911.
If you see a ,D. at the end of a name, that means Lady of a female and Lord for a male.