"These robots-- they're like a window to my past."
"Whoopee. Roots belong underground, where you don't have to see them."
--Zeta and Ro, Change of Heart
OKAY SO THAT QUOTE IS A TOTAL LIE.
Thus, it is an awesome quote to start this essay with. Contrary to what she says there, family is pretty damned important to Ro. (She spent most of the episode protesting too much, of course.) And after I had her laptop spit out that bit of personal info about her sans context, I realized that while it wasn't a big thing and she knew it, she was still offended. Ro has some pretty strong ideas of what family ought to be. She went for years without anyone who she could really consider family, be it blood related or not, so after meeting Zeta she can't imagine how anyone could dismiss family so easily. And I can't blame her, since her perception of family comes from her relationship with Zee:
you take care of them, and they take care of you. Period.
Zeta is Ro's family throughout the series: she tells him as much in the first season. He was the first person in years to genuinely care for her, to make sure she had food and shelter and was safe and happy-- and not merely because he was obligated to. As much as his selflessness and naivete makes her facepalm, that's what Ro loves about him. She only yells and/or facepalms because she wants to protect him. It's part of the reason she takes Agent Bennet's single-minded pursuit of Zeta almost personally-- despite Zeta obviously having so much respect for life and how much Zeta's risked his life to save others, Bennet rewards him for it by ignoring it in favor of his suspicions. True, Zeta spoils her rotten and Ro doesn't stop him, but... she adores him and will do anything to protect him. That's what families do. They take care of each other.
So this is the kind of relationship Ro thinks of when she thinks of "family." Naturally, she's got a few strong notions about what family should mean. You don't give up on family, you don't forget them, you look out for them, and you don't sell them out to the NSA. Which, if you ask me, is why she gave her biological brother Casey so much slack when they finally met in "On the Wire." This episode is important because I pulled Ro right after that episode in canon.
Sure, Ro was cautious about the meet-up, and she wasn't stupid about it. But unlike her ex-foster family, Ro trusted Casey with the whole story about Zeta right off the bat. The whole story. Maybe it was because she learned her lesson trying to deceive the Morgans, but I think it was also because she was pretty sure Casey was her brother. This is pretty damn significant given how reluctant Ro is to trust people.
Then, Casey led them into what looked like an obvious trap, the deal being that if Bennet got Zeta he'll let Ro off the hook. Ro, of course, didn't react well, screaming that they were family and you do not sell family out. So in the middle of that setup, Casey reached for Ro, who was already ticked off, and told her to "trust me."
...And with only that to go on, she did. Because, again, he was family. And Casey "accidentally" piloted the chopper in Bennet's way, allowing Ro and Zee to escape. Ro's smug reaction? "He's a Rowen." (In spite of Casey taking his foster mother's name, something that irked Ro when she found out.) For Ro to acknowledge him as family, not once but twice, Casey basically blew off a career in vid news and risked the NSA's wrath in order to let Ro and Zee escape. That was not a small thing to do. I kind of think the acknowledgement was also due to the sneaky way he pulled it off, but it had more to do with the fact that he would do that for her in the first place. To Ro, that's what family is about. It's what it should be.
So when the computer blabbed to the world at large that she'd forgotten she had a brother with no context, Ro was offended. Sure, the circumstances were understandable; they were split up when she was too young to remember, and she lost the only photo she had of him to her thieving foster sister when she was 8. Seven years bouncing around the system with no reminders of her brother later, it's understandable that she would forget. Ro doesn't entirely feel that way; her astonishment when Tiffy gave her the photo back wasn't merely due to the fact that she had a brother, but also that she forgot.
She didn't beat herself up over it. Ro doesn't do that. Her reaction was more like "wtf I have a brother HOW THE HECK DO YOU FORGET SOMETHING LIKE THAT?" And after having met Casey, who definitely stood up for her the way family should, who didn't forget, that reminder hit a nerve. She knows it wasn't her fault, and she's not going to emo about it, but family is important to her and Camp's computers don't know what the hell they're talking about.