Is it stated why Bariss chose Ahsoka to frame for Bombing of the Jedi Temple Hangar?
To turn her against the Order
Ahsoka Tano believed that Barriss was trying to convince Ahsoka to join her cause (i.e., opposing what she believed the Jedi Order to have become). Though we don’t see this in an episode, it’s made clear in the canon novel Ahsoka:
Barriss had been angry with the Jedi Order and had sought to win Ahsoka’s sympathies, if not her outright alliance, but she’d done so in the cruelest way imaginable: by tampering with Ahsoka’s own choices. To have a person she considered a friend use her to unleash such deep anger and channel it at the Order had changed every part of Ahsoka’s outlook.
Presumably Barriss confessed her motivations to Ahsoka after the events of “The Wrong Jedi.”
While it’s clear enough from this what Barriss intended, it’s not obvious how framing Ahsoka helped her achieve this. Given that she did her best to pin the blame on Ahsoka, it seems likely that her goal was to get Ahsoka to see the Jedi Order the way she did, by personally showing her how far it had fallen. It would seem likely that by framing Ahsoka, Barriss intended to make her angry at the Jedi Order, and perhaps later seek her assistance in working against it (while presumably still concealing her own role in the bombing).
Of course, she had to set Ahsoka up to take the blame first, so her own role wouldn’t be suspected. It would seem that she was trying to kill two birds with one stone: by framing Ahsoka she (a) pinned the blame for the bombing on a convenient target, and (b) hoped to make her amenable to seeing her side of things (“See how quickly they turned on you?”). A former Jedi locked in Republic prison might have been quite vulnerable to insinuations of Jedi corruption.
I suppose she wasn’t counting on Tarkin seeking the death penalty, since this would have obviated any plans to capitalize on Ahsoka’s anger. Indeed, perhaps part of the reason she confessed was to save Ahsoka.
It seems quite likely that this explanation or something similar was intended, at one point, to be included in a future episode of The Clone Wars. According to Filoni:
Q: What happened to Barriss Offee?
A: In the original script, she died. She blows up her own prison cell. “I didn’t want that fate for Barriss, so I cut that part. I had plans for Barriss.”