explanation of “our house’s rat”
This joke doesn’t seem funny in English at all, and after thinking about it for quite a while, I’m not sure there is any way to both properly translate it while keeping it funny. Translating it any way that allows the reader to quickly understand also cuts down the humor simply because nobody likes having a joke spelled out for them (which is also why our explanatory posts are rarely funny).
Getting back to the joke itself, all you have to keep in mind is that the 药 in 老鼠药, on its own is usually translated as “medicine”. The child, not knowing that 老鼠药 is actually rat poison, assumes that the mother is trying to cure the rat of some disease by giving it this “medicine”.