Honestly, a LOT depends on the tone of the conversation and who's talking.
Is the character being dismissive/scornful? ("What are you going to do, throw me in a madhouse?") Is she afraid? ("You aren't going to put me away, are you?") I feel like the more scornful she is, the more likely she is to actually name the place, whereas the more fearful she is, the more likely she is to talk around it, unless her fear tends to make her angry and that anger makes her reckless, and then we might be back to naming it outright.
Also important: is this a "polite" conversation? Is this woman well-bred/genteel? The more "polite" the conversation, the less likely they are to name the thing out loud. There was a major stigma against mental illness. If you could talk around it instead of about it, you would.
Here's an example. According to Elizabeth Keckley, Mrs. Lincoln's dressmaker, after Willie Lincoln died, Mary was in a desperate state and Mr. Lincoln said to her, "Mother, do you see that large white building on the hill yonder? Try and control your grief, or it will drive mad, and we may have to send you there."
In this case, he's talking around it to a certain degree (or at least Mrs. Keckley has him doing so).
I will push back against lonestarlibrarian a bit. Mental asylums could be awful places, and in fact I would say that the wealthier the family, the less likely they would be to send their loved one there (presuming the family doesn't want to get rid of this person and wants the best for him/her). Wealthier people could afford to care for this person in their home, either doing it themselves or hiring someone to assist. Less wealthy people might have little choice but to put this person in an asylum (because they don't have time/money to care for the person themselves). Unfortunately, the treatment was likely to be terrible there. Of course, the in-home care could be poor/awful, too, and the mentally ill person is likely to have been hidden away, and there were decent institutions (but not many). So there are a lot of variables here. The best option for someone suffering mental illness was probably to be cared for at home by a loving family with resources. The worst would be to be dumped in an abusive institution and forgotten.
In any case . . . "madhouse" seems like a good option!