Coming to you straight from the Bible, Inherit the Wind is a shout-out to the book of Proverbs. A verse, which is quoted twice in the play, reads, "He that troubleth his own house shall inherit the wind" (Proverbs 11:29).
This passage refers to the idea that someone who causes problems for mom and pop is going to get a whole lotta nothing come inheritance time. In the play, Brady warns Brown against damning his daughter, citing this verse.
Hornbeck and Drummond use that same verse as an obituary for Brady after he dies, probably because he got himself so worked up that he ended up with nothing… even after his courtroom win.
Since Bert ends up with the moral victory, even though he loses his case, it's probably not him that's troubling his house. In fact, the one who ends up with nothing is definitely Brady: no presidency, no chance to talk on the radio, no riding out of the courthouse on everyone's shoulders.
But why would Brady be troubling his house? Well, maybe the fact that he uses the justice system for personal gain rather than to seek justice is a way of showing that he is troubling the house of his nation. That's one way to think about it.
Also, Reverend Jeremiah seems to think that Rachel is the one doing the troubling. But even Brady recognizes that it's the Rev who's out of line. He ends up losing his daughter (to Bert), and since we know he has no wife, we believe he ends up all alone. That's what you get when you talk about how your family members are going to burn in hell, we guess.