Good question! 

Well, if we’re going by medieval England as we usually are, you can pick a last name whenever there’s enough people in a given area that it’s tricky to keep track of who’s related to who, because last names weren’t a signifier of status as much as titles or coats of arms or mottos (which required approval from above and usually some payment). So Ralph the spicer becomes Ralph Spicer and becomes considered a particularly well-established member of the merchant classes who’s following the forms. 

The big change is what happens when Ralph Spicer wants to make the transition from merchant to the nobility and found House Spicer. This was a difficult and slippery process, because one would have to start by becoming a gentleman (which generally required that one owned a manor that could support you without your own labor), then ascend to the status of esquire (which definitely required approval from above in the form of achieving some form of office that brought the title of esquire with it, usually Justice of the Peace or Sherriff or something else having to do with the law), then become a knight (which requires being knighted), and generally only after could one aspire to the nobility. Along the way, there were not merely legal forms one had to pass through but cultural forms as well – gentlemen were supposed to learn to be “genteel,” to get into chivalry and noble sports and out of trade, they had to get a coat of arms which meant their pedigrees had to pass muster with the College of Heralds (although this could often be finessed with the right payments to the King and then to the herald). 

So if Ralph Spicer was in England, he would probably have started by getting his hands on enough land that he could pretend that he wasn’t a spice merchant any more or to give it up altogether, then gotten himself a Justiceship by knowing the right people or bribing people, then making sure that his kids were squiring for a local knight and then marrying them off to any impoverished nobility in the area, and then making a bid for a knighthood and hope to ascend from there.

But Westeros is less legalistic and bureaucratic than that, at least as how GRRM describes it. It could be as easy as Ralph getting his hand on a bunch of land then paying off a hedge knight to make all of his sons knights.