When Barry, Cisco, and Caitlin first finds Gideon, she tells them about Barry saying "Founding member of..." but is interrupted by Barry. What exactly will Barry be the founding member of?
Most probably the Justice League.
From the preview comic for Episode 20 of Season 1 of the Flash, The Trap on the CW's twitter, they reveal that Gideon was cut-off after she said he was the" Founding member of the Justice..."
Whereas Batman aka Bruce Wayne is the founder of the Justice League in the DCEU, Barry is the founder of the Justice League in the Arrowverse. In fact, Barry has been the one behind at least two or three Arrowverse crossovers - as far as I know. He was the one who arranged for both Team Flash and Team Arrow to battle Vandal Savage in the fall of 2015. He was the one who arranged for Team Flash, Team Arrow, the Legends and Supergirl to form a team and battle the Dominators' invasion in the fall of 2016. It was his wedding to Iris West that led to the gathering of the Arrowverse heroes to battle the Earth X villains in the fall of 2017. And in the end it was Barry who literally formed the Justice League following Crisis, using STAR Labs' hangar as the League's future headquarters.