While a literal meaning for the selection of the name is not something Saki ever explains in detail, the reader can make several connections, drawn by a combination of a play on sounds, words, and traits that help further distinguish the Framton Nuttel's character traits.
First, we must note that there is no physical description of Framton, except that he suffers from a nervous condition that renders him awkward and unable to conduct himself with the social decorum expected by one who is going to ask for an act of kindness from a stranger. He is odd from the very start, when he "endeavoured to say the correct something which would flatter the niece of the moment without discounting the aunt that was to come."
Based on just that, we can say that the man is frazzled enough to go on a frenzy at any moment. We could assume that words such as frenzied, frazzled, and frumpy are backdrops to the name "Frampton."
As far as the last name "Nuttel" goes, it is generally accepted that Saki uses the root "nut" in the form of an adjective that may help describe Framton as someone "nutty," or about to go crazy.
The etymology of the word "crazy" dates back to 1846, 24 years before Saki's birth in 1870. The the earlier version of the word was the 1785 word "nutt," as in the phrase "to be nutts upon," which meant "to be fond of" something. The phrase to be "off one's nut" (head) itself dates back to 1846, so Saki was amply exposed to that terminology prior to the start of his writing career.
Conclusively, Framton Nuttel's strange name is meant to label him as a strange man. Whether the name Frampton is allusive to the man's frazzled state of mind, the last name "Nuttel" is more than likely used on purpose to definitely describe Framton as a true "nutcase."