Gideon posseses the folowing abilities, though for some ease (and fun) of play I'll be changing them slightly.
SUPERCOMPUTER. Gideon has full control over the Waverider and her systems and frequently pilots and time jumps the ship. She also has a vast knowledge of the timeline and is capable of detecting time aberrations, including predicting how altering past events may affect the future. In addition to operating the navigation of the Waverider Gideon also administers medical treatment, including healing crew members and administering sedatives. She has also guided Martin Stein in how to perform brain surgery.
Since Gideon has been placed inside an android body snd is no longer connected to the Waverider her knowledge will be limited. She still has supercomputer processing, but not all of the databases and information that she previously had access to. The more immediate/skills things she'll retain, e.g. mefical knowledge, but not every detail of history, e.g. biographical details of evry prrson to exist.
TELEPATHY. Gideon has shown that she is able to see into the team's dreams, and knew the state of Rip's mind prior to entering it, able to advise Sara and Jefferson on how to help save him.
Though Gideon will still be able to do this she won't automatically do it to people that she doesn't knoe. Her focus will still be on her crew and ensuring their safety.
HACKING. Gideon is able to sabotage other AI systems and can mask her data trail to appear as a different source.
Given that the network is magic she might encounter some difficulties when trying to hack it.