Stinson Out is something Barney Stinson, and possibily James Stinson, say when they are leaving, and want to make sure everybody notices.
Examples[]
In Robin 101, Ted tries to convince Barney to open up to Robin emotionally and 'try a little harder'. Barney refuses saying "only thing - and this is just me - I like my testicles attached to my body, rather than rolling around next to some eyeliner in Robin's purse. Stinson Out!". However, after a few seconds, he claims 'Stinson back in!' wanting to hear Ted's ideas.
Barney used this in The Poker Game, when William Zabka had returned the photo Ted Mosby had 'destroyed'. His exact words where: Billy rules, Ted drools, Stinson Out.
He also used it in The End of the Aisle, when Marshall and Lily realized they had broken most of their wedding vows, Barney leaves the room screaming 'Stinson Out'.