I've just started watching Legends of Tomorrow, and the time vessel's captain is aided by an AI called Gideon. Is that the same or different Gideon from the one in The Flash? Also, is the name a reference to Gideon?
Gideon is an interactive artificial consciousness.
Now, consider this scenario: You have an iPhone. Your friend also has an iPhone. Both of you use Siri on your phones.
Think of Gideon as Siri, from the Future. A lot of people probably use it.
The Gideon in S.T.A.R. Labs had a different voice than the one on board the Wave Rider. That's because they changed the voice actor due to a scheduling conflict.
It is however interesting to note that (in the context of canon) Rip Hunter's Gideon has a British accent, and since he is British, we can assume that Gideon can be programmed to speak with different accents. So his Gideon has that accent.
So, is it Gideon? Yes.
Is it the same Gideon from S.T.A.R. Labs that the Reverse Flash / Eobard Thawne / Professor Zoom had? No.
How about this? The Gideon on the Wave Rider is the future version of the Gideon on the Flash, which is a future version of the Gideon that Thawne stole from the Wave Rider. Kind of like when in the episode of Dr. Who where there was 3 Dr.'s. The oldest one programmed his sonic screw driver and the answer came up on the newest Dr.'s screw driver because it had calculated through all those years. Or not.
I believe it is the same gideon becouse in S2 thorn in on the ship looking for something.
I think he finds gideon and copies her base code and her info about the time line, but at the end of S2 thorn is killed so I think he makes a time remnet and gives sead remnet his gideon and leaves it for thorn in the flash.