The Hornets announced that, following a trial spell, the 22-year-old has signed a six-month contract with the option of a further deal, subject to international clearance.
Midfielder Mitchel will continue his development with Watford's under-23 side having opted to move to England after nine years at Dutch side Almere City.
Dad Dennis won the Premier League three times during his 11-year stint at Arsenal, while he sits fourth on the Netherlands' all-time scoring list with 37 goals in 79 international appearances.
Mitchel admits he has become accustomed to comparisons with his famous father but is now plotting his own path in the professional game.
"When I was younger it was difficult [to be compared all the time] but I've adapted to it now," Mitchel told Watford's official website.
"It's just my name and I'm my own player and hopefully I can leave my own legacy. He's been amazing to me and for me. He's my number one fan and has always been there for me.
"It's a privilege to have him as my dad and to work with him on my game."
Watford have also signed Maurizio Pochettino, son of Paris Saint-Germain boss Mauricio, during a busy transfer window as they look to seal a quick return to the Premier League.
Dan Gosling is another new arrival, while Glenn Murray's season-long loan from Brighton and Hove Albion has been terminated.