There are apparently two quite separate origins, although with curious similarities which suggest a possible common source in pre-history. The first is from the ancient Gaelic-Breton compound "Rhyd-derch" which translates as "famous chief", a meaning which no doubt contributed to its early popularity. In the Middle Ages a secondary meaning was "the red haired one", although this in itself may also have harked back to the original meaning. The second origin is Germanic, from the pre 5th century, when German tribes, particularly the Vizigoths swept down into Spain and Portugal. They left behind many examples of their names, of which this is one of the most popular. The derivation is from 'hrod' meaning renown and 'ric' - power, a not dissimilar translation to the Gaelic. The Germanic origin: HROD-RIC . HROD means GLORY; it is found in many other names, like HDROD-WULF, which gave RODOLPHE or RUDOLF. RIC means WEALTH or POWER; our modern word RICH would be derived from it (in the Frankish language, RIKI meant POWERFUL). So the Rodrigue would be of Germanic descent. As the surname is very widely used in the Iberian Peninsula, one might reason that it would have been introduced by the Wisigiths, a Germanic people which settled in Spain and in the South of France from the VIth Century. Besides, one of their kings (in fact, the last one) 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. The name was at first baptismal and pagan, although at later times after the 7th century it became closely associated with the early Christians. The RODRIGUEZ surname is very common in the Iberian world, as we have seen above; it has also been adopted among the Jewish population of the peninsula. Because of the Inquisition which was conducted there after the Reconquista from the Arabs, many Spanish Jews had to leave their country; so we can find in France, at that period, some Jewish families named RODRIGUEZ. Some have even gone further: there are Turkish RODRIGUE (and yes, without the final z) of Jewish origin, whose ancestors settled in Turkey as early as the XVth Century, and who are now living in Ankara.
The first authenticated RODRIGUEZ surname recording is believed to be that of Rhodri Mawr, or Roderic the Great, King of Wales, who died in 877 a.d., however this was not a surname. Early examples of these taken from church and civil registers include Christobal Rodriguez de Leon in 1536, and Juan Rodriguez de Santos, in 1662, at Valladolid, Spain. The coat of arms has the bl;azon of quarterly, red and gold, in one and four an eagle displayed in gold, in two and three, three fleur de lis in blue. The first recorded spelling of the family name is shown to be that of Heinrich Rodigerus, which was dated 1260, in the charters of the city of Lubeck, Germany, during the reign of Emperor Alonso X of the German Empire, 1257 - 1273. Surnames became necessary when governments introduced personal taxation. In England this was known as Poll Tax. Throughout the centuries, surnames in every country have continued to "develop" often leading to astonishing variants of the original spelling In the 2000 census, Rodriguez was the ninth most common surname in the United States, making it probably the first time in U.S. history that a non-Anglo name ranked among the 10 most common surnames (the Hispanic surname Garcia also cracked the top 10 at #8).
Comment: This is very interesting. I would like to add that the surname "Raderootjes" which seemingly comes from the Netherlands, might also a variant of the surname "Rodriquez" in my humble opinion.