There are only a few comic characters of the Silver Age who qualify as "franchises". That is, they were so successful that they spawned spin-off series that were successful in their own right.
One of those characters was the mightiest human being who ever lived. Another was the lowliest private in the Army. A third was a freckle-faced teenager. And the last was "a poor little rich boy". It's kind of an odd mixture; two people who lived lives we could never dream of, and two "everyman" types.
Archie was the undeniable comics franchise of the Silver Age, producing a staggering number of spinoff mags: Little Archie, Archie & Me, Archie's Madhouse, Archie's TV Laugh-Out, Betty & Me, Everything's Archie, Jughead, Jughead's Jokes, Life with Archie, Reggie & Me, Reggie's Wise Guy Jokes, in addition to the Archie comics that started in the Golden Age, like Archie, Betty & Veronica, Pep, Laugh, etc.
Superman, of course, had been a franchise character almost since his introduction, spawning his own mag, World's Finest, and Jimmy Olsen and Superboy in the Golden Age, while that latter character took over the lead in Adventure Comics. He added Lois Lane to his regular appearances, and was the designated character to introduce DC's line of Annuals. While he did not generate as many new titles as the other characters in this post, what he did generate sold amazingly well: In 1962 the top four titles in sales were Superman, Superboy, Lois Lane and Jimmy Olsen, and the only Superman-related title to finish out of the top ten was Justice League of America (which was 11th, and generally did not feature Superman on the cover or in the story at the time). In 1965, the top six titles in sales were all Superman-related.
Marvel did not have a franchise character/feature in the Silver Age, although they had two that would eventually earn that designation--Spiderman and X-Men would eventually start branching out into more and more books. You can make a teeny-tiny argument for Millie the Model, but I'd rather not.
Harvey was the only publisher to have two franchises: Richie Rich and Sad Sack. Richie Rich's own book only launched in 1960 (he started in 1954 as a backup in Little Dot), but within a few years he had launched Richie Rich Dollars & Cents, Richie Rich Millions and Richie Rich Success Stories. In the 1970s the number of Richie Rich titles continued to spiral upwards as every valuable item in the world from Gems to Gold and Silver became the inspiration for another magazine.
Sad Sack was similarly successful in launching new titles. Sad Sack & the Sarge, Sad Sack Laugh Special, Sad Sack's Funny Friends, Sad Sack's Army Life Parade and Sad Sack World all had long runs starting in the Silver Age.
Any others? Donald Duck can make an argument, but Uncle Scrooge was a well-established magazine before the Silver Age started, and the Beagle Boys and Gyro Gearloose didn't last long enough to qualify. Batman, like Spiderman and X-Men was a franchise feature waiting to happen; I don't think you can count his late and tentative (at the time) takeover of the Brave and the Bold.
Update: Snard points out a good one in the comments: Casper the Friendly Ghost. Casper had his own book and spun off long-running titles like Spooky, Wendy the Good Little Witch, Casper and Nightmare, Casper Ghostland.