Had two different ideas, decided to combine them:
1) Make Peter Parker a "legacy" hero like Hank Pym's Ant-Man. He had his adventures, but he was never in the spotlight, and eventually he had to retire, leaving Spiderman to become a bit of an urban legend, if a popular one. Maybe have Spiderman referenced in a couple of the Netflix series, since they're all set in or around Spidey's territory, to help build the lore, and maybe even have an older Parker himself make a token appearance.
2) Make the "new" Spiderman an Inhuman, because that's the current way to get away with not explaining a hero's powers since mutants aren't on the table and because it opens up the Inhumans movie as a convenient way to introduce him; I'm still betting that movie covers the Inhumanity story line from the comics at least by the end of it, releasing the Terrigen Mists across the world and creating thousands of new "capes." A few of those could take up the mantles of heroes, past or present: Kamala Khan as Miss Marvel...Miles Morales as Spiderman. Could do a small scene with Peter Parker tracking down the new Spiderman and offering to show him the ropes or something. Then bam, Spiderman for Infinity War part 2!
(I really just want Marvel to use the Miles Morales Spiderman: if nothing else, it would be a good opportunity to tell a different story.)