A Song of Ice and Fire is all about subverting tropes, and Catelyn's story is no exception. Rather than focussing on Robb, the heroic young king, we see all of his battles, his successes and his failures, from the point of view of his mother, a sensible, compassionate woman who just wants everyone to grow up and go home before they all end up dead. She's not focussed on vengeance for her husband's death, or ambition for her son, but neither is she purely a traditional fantasy mother figure, someone who steps back and worries for her sons and plays no part in events. She is an important figure, who is involved in politics, who occasionally makes mistakes, but who is generally sensible and under-appreciated by those around her.
And, of course, many people hate Catelyn for it. She's hated because she is flawed and makes mistakes -- something a mother figure should never do. She's hated because she doesn't treat Jon Snow the same as her own children, because she interferes in politics (and is usually right), because she acted against the Lannisters instead of trusting them and hoping for the best (something that ultimately got her husband killed). In my opinion, these things all make her a more interesting, realistic character, but according to others, she shouldn't be interesting or realistic. She should be perfect, she should be out of the way, or she should leave.
Unfortunately, the TV show seems to support this opinion.
The addition of Catelyn's early resentment of Jon Snow isn't necessarily a terrible thing. Considering that her new husband, who she barely knew, disappeared off to war and came back with a son, it's unsurprising that she wasn't thrilled with Jon Snow's presence, and hoped that he would disappear, without really meaning what that would entail. And it's not problematic that she cared for Jon Snow when he was ill and prayed that he would recover, as Catelyn is generally a caring woman, and, despite her refusal to treat Jon Snow exactly the same as her own children, she's not someone who likes to see other people suffer.
But this history was clearly added as a way to put Catelyn in her place. She did not just feel resentment for this child, as she does in the books, but prayed to the Gods that he would die. It's explicit, and it's extremely out of character for a woman who doesn't even want to see her enemies die. Especially when she wasn't motivated by discomfort with her husband, or uncertainty over her own son's future, but by jealousy for a woman she has never and will never meet. She is, as she says herself, "the worst woman who ever lived" for wishing death on this boy -- and it is a pretty terrible thing to do. The writers of the show have transformed Catelyn's dislike for Jon Snow, and her one-time grief-fueled declaration that he should have fallen instead of Bran, into a passionate, almost murderous kind of hatred. For if she does not love Jon Snow enough, she must be that terrible. It is an either-or proposition.
And in order to make up for her horrific behavior, she must swing far in the other direction, and treat him exactly the same as her own children, including giving him the name "Stark." And we get the sense that this would be the normal thing to do, that this would be the thing any caring person would do, rather than an extreme. This is proper behavior, how she should have acted from the beginning, despite the fact that she isn't his mother, has no obligation to be his mother, and that giving him the name "Stark" breaks all expectations in Westeros and could potentially threaten her own children's inheritance. But motherhood, to the writers, seems like a binary system. Either she's perfectly selfless and loving to all children, no matter who they are, or she is a horrid woman who deserves misery. No other options exists.
So when she says that she couldn't keep that promise, it isn't a declaration that this was something said in desperation, something unrealistic. It is treated as a sign that Catelyn is in fact the "worst woman who ever lived." And then Catelyn, intelligent, compassionate Catelyn, takes entire responsibility for all the horrors that have come to her family -- Ned's beheading, Sansa's imprisonment, Bran and Rickon's deaths. All because she did not love Jon Snow. And no one corrects her. She makes that declaration, and the scene ends, because although assumedly the viewers aren't meant to agree that this is all her fault, we are meant to find her grief and regret reasonable. We're meant to agree that she is not a good person, and consider her regret and self-loathing as a kind of redemption. If she sees how awful she's been, and despises herself for it, then perhaps she can be an acceptable character. Otherwise, she deserves all the hate she receives.
It is complete character assassination, disguised in an attempt to redeem her. And that's not the kind of "development" of female characters we need to see.