By 1962, with the success of the Fantastic Four and other characters, Marvel editor and head writer Stan Lee was casting about for a new superhero idea.
In Stan's autobiography, he cites the non-superhuman pulp magazine crime fighter The Spider as an influence and both there and in a multitude of print and video interviews said he was inspired by seeing a fly climb up a wall — adding in his autobiography that he has told that story so often he has become unsure of whether or not it is true, but it seems to be good enough for the comic community.
Artist Steve Ditko, in a 1990 article by himself, gave a more prosaic origin story for the name:
"In a discussion with me about Spider-Man, Stan said he liked the name Hawkman but DC had the name and character. Marvel would add Ant-Man [and the Wasp] so it would have the insect category. (Technically a spider is not an insect). From that I believed Stan had named the character.
Stan approached Marvel publisher Martin Goodman to seek approval for his new Spider-Man character. In a 1986 interview, he described in detail his arguments to overcome Goodman's objections. Goodman agreed to let Stan try out Spider-Man in the upcoming final issue of the canceled science-fiction/supernatural anthology series Amazing Adult Fantasy, which was renamed Amazing Fantasy for that single issue, #15 (Aug. 1962) which was... Amazing Fantasy #15 what every geeky boy/man ever dreamed of having.
In a 1982 interview, Jack Kirby claimed Lee had minimal involvement in the character's creation, and that it had originated with Kirby and Joe Simon, who in the 1950s had proposed a character called The Silver Spider for the Crestwood comic Black Magic until the publisher went out of business. Unfortunately, as much as all of us love Jack Kirby we think 7 years is too long a time to be able to claim an original idea as yours when someone else thinks it up as well. It was only a matter of time before someone else thought up the same idea.
Simon, in his 1990 autobiography, disputes Kirby's account, asserting that the supernatural anthology Black Magic was not a factor, and that he (Simon) devised the name "Spiderman" (later changed to "The Silver Spider"), while Kirby outlined the character's story and powers. Simon later elaborated that his and Kirby's character conception became the basis for Simon's Archie Comics superhero The Fly, introduced in early 1959.
... well, there you have it! .. Now go buy our Spiderman t-shirts and check out the rest of our Spiderman merchandise.