Claydon, 30, took 22 wickets during a three-month loan spell at Canterbury this season.
Griffiths, 27, came through the academy at Hants and made his debut in 2006.
Meanwhile, Kent youngsters Fabian Cowdrey, Daniel Bell-Drummond, Calum Haggett and Sam Northeast have all penned new deals with the county.
Kent chairman of cricket Graham Johnson described the signings as "extremely exciting".
"All are ambitious players who seek a positive environment in which to develop their talent," he said. "They are also ambitious for the club and keen to play their full part in all aspects of developing a winning culture.
"Within the group there is a mix of youth and experience, with players coming through the Kent system and others seeing Kent as the place to fulfil ambitions - a blend that is essential in a squad.
"We continue to look at the balance of our squad as there remain players who are out of contract at the end of this season and discussions continue with them."
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