The former Tranmere shot-stopper attracted interest from a number of clubs this summer, but opted to join the Britannia Stadium outfit.
Simonsen arrived at Goodison Park for £3.3 million in the summer of 1997, but failed to have the desired impact with The Toffees.
The 25-year-old is now hoping a change of clubs can resurrect his career and he can force his way back into Sven Goran Eriksson's England fold.
"Steve's delighted to have sorted his future. He was looking for a fresh start, that's why he turned down Everton's offer," said Simonsen's agent Lee Payne.
"There were a number of clubs interested in signing him, but once he saw the set-up at Stoke he knew that's where he wanted to go.
"He's only 25 and he hopes that he can prove himself again and get back into the England set-up as he is very ambitious."
