Framton Nuttel certainly is an unusual name. It helps to characterize the visitor as an odd and eccentric man. The last name of Nuttel must suggest to most readers that the visitor is a little bit nutty, just as Aunt Sappleton's last name suggests the woman is a bit of a sap. Saki may have had another reason for giving the visitor such an unusual name. Authors often have problems choosing names for their characters. If they choose a common name, such as "Bob Wilson," a number of male readers who happen to have that name (or people who know a "Bob Wilson") might think the author is writing about them! This is not important if the character is fairly innocuous. Framton Nuttel is not innocuous, though; he is depicted as an object of ridicule, so Saki probably gave extra thought to Nuttel's name and came up with a name that would keep Nuttel from being mistaken for any man in the general population.
As another example, Vladimir Nabokov created an outrageous character in his novel Lolita. Nabokov didn't want any living man to be offended, so he chose Humbet Humbert, a name that seems as weird as the protagonist and would be impossible to find duplicated in any phone book.