Robert was getting the strong wine regardless of what happened, it was just a matter of time. Ned could've stayed at Winterfell and the whole story could've been about them training their direwolves and Cersei would've still strong wined Robert. She hated him and everyone knew it.
Taking Tyrion was smart for multiple reasons. She has every reason to think he tried to kill her son at least once, probably twice, so she's allowed to ask for some kind of justice in the eyes of the gods and the people. She is sneaking back to Winterfell in disguise after going to King's Landing to plot out potential war strategy with Ned and he is about to start loudly asking why she's not in Winterfell taking care of Bran. She has a letter from her crazy ass sister that's says the Lannisters are actively offing people like Jon Arryn, which (if true) would mean there is some massive reaching, treasonous plot getting acted out by the Lannisters.
So she's sitting there thinking who and how she can rally support from the Riverlands if the Lannisters don't listen to Ned/Robert/Littlefinger and bam - a Lannister - and not just any Lannister but the Lannister she has a right to take into possession and ask to face justice for the accusations against him. The Lannister that was about to start loudly asking why she was traveling around in disguise, which looks shady as fukk. Ned has drilled into her head that he can handle Robert because they are homies from way back and under all that drunken fat, he's a decent person that would look at things fairly and do the right thing. If Ned had been right about being able to handle Robert, which she believed to be true, she was grabbing up a primo bargaining chip for them to help the cause and keep things from getting too out-of-control.
Where she really fukked up was going to the Vale and her sister. If she had gone to Riverrun and kept the crazy/Littlefinger out of the mix, she'd have had Tyrion which would've given massive support to the Ned/Robert cause she believed to be happening in King's Landing. Unfortunately, none of that was true and happened. Instead Littlefinger was fukking her sister and Robert wasn't about to start pushing the Lannister's in any kind of real way, regardless of what Ned thought.
Taking him was smart and then acting like she was taking him to Winterfell was smart, everything after... not so smart.