Origin of Rodrigue Surname
According to the commercial website The House of Names, the Rodrigue surname is first found in Brittany.
Real Rodrigue of the Rodrigue Family Association, however, writes that the surname has a Germanic origin, introduced by the Wisigiths, a Germanic people who settled in Spain and in the South of France from the VIth Century. Their last king was named RODERIC. With the latinization of the Wisigoths, the name RODERIC became RODERIGO, and then RODRIGO. From this forename came the Spanish surname RODRIGUEZ (son of RODRIGO), which was slightly altered to become RODRIGUES in Portugal.
Origin of Rodrigue in North America
Real Rodrigue has identified six different lineages of Rodrigues in North America: four Portuguese and one Spanish. His website gives details of each of those lineages. 
References & Footnotes