Quote from Wikipedia, which incidentally has an excellent article covering this controversy:
In addition to the two documents cited, there are others that confirm the identification of the Genoese Christopher Columbus, son of Domenico, with the admiral of Spain. An act dated 11 October 1496 says:[7]
Giovanni Colombo of Quinto, Matteo Colombo and Amighetto Colombo, brothers of the late Antonio, in full understanding and knowledge that said Giovanni must go to Spain to see M. Christopher Columbus, Admiral of the King of Spain, and that any expenses that said Giovanni must make in order to see said M. Christopher must be paid by all three of the aforementioned brothers, each one to pay a third … and to this they hereby agree.
In a fourth notarial act, drawn in Savona on 8 April 1500, Sebastiano Cuneo, heir by half to his father Corrado, requested that Christopher and Giacomo (called Diego), the sons and heirs of Domenico Colombo, be summoned to court and sentenced to pay the price for two lands located in Legine. This document confirms Christoforo and Diego’s absence from the Republic of Genoa with these exact words: “dicti conventi sunt absentes ultra Pisas et Niciam.”[nb 5]
A fifth notarial act, drawn in Savona on 26 January 1501, is more explicit. A group of Genoese citizens, under oath, said and say, together and separately and in every more valid manner and guise, that the Christopher, Bartholomew and Giacomo Columbus, sons and heirs of the aforementioned Domenico, their father, have for a long time been absent from the city and the jurisdiction of Savona, as well as Pisa and Nice in Provence, and that they reside in the area of Spain, as was and is well known.
Finally, there is a very important sixth document from the notary of Bartolomeo Oddino, drawn in Savona on 30 March 1515. With this notarial act, Leon Pancaldo, the well-known Savonese who would become one of the pilots for Magellan’s voyage, sends his own father-in-law in his place as procurator for Diego Columbus, son of Admiral Christopher Columbus. The document demonstrates how the ties, in part economic, of the discoverer’s family with Savona survived even his death.”
His son wrote: Colombo … was really the name of his ancestors. But he changed it in order to make it conform to the language of the country in which he came to reside and raise a new estate …
I hope that wasn’t too long, and that interested parties read the entire Wikipedia article, it’s extremely well researched.