It wholly depends on the Chapter. First thing you have to remember, humans no longer have names we would reconize. It is 40,000 years in the future. As well, any linguistic would tell you, coming into contact with so many races and lost branches of humanity changes language. That said, here's some examples:
In Dawn of War Omnibus, a guardsman who gets recruited by the Blood Ravens, Ckrius, keeps his name. Its also hinted that Captain Gabriel and Librarian Isador kept their names as well.
In the Space Wolves, you lose your name, until you earn it back by becoming a Grey Hunter or other promotion.
Certain Chaos Legions, like the Thousand Sons, have maintained the same names since first leaving Terra with the Emperor. Ahriman is one of them. His full name always being Ahzek Ahriman.
It would note, however, that most of the named Space Marine characters in novels have names that are closley associated with their Chapter. This is true in particular for those chapters with am angelic feel: Blood Angels (Dante, Memphiston), Dark Angels (Sammel, Belial, Ezekiel). That may just be designers ease of name, or that the chapter renames its battle brothers something more in line with the Chapters background.
"Take a moment, please. Ponder the depths of your insignificance."
"It would be disappointing if you were unable to overcome this simplest of defenses. Hilarious, but disappointing."
Crimson Fists (C:SM)
WARMAHORDES (Skorne, Blindwater Congregation, Retribution of Scyrah)
DROPZONE COMMANDER (PHR)