What are the limits of the word "invention" (see details)?

Inventions are often contrasted with discoveries. The former are things that we bring into existence, whereas the latter are things we find that already exist. This can’t be all there is to the distinction, however, as we don’t say that we invent children, even though they do not exist before we create them.

The real inspiration for this question, though, is an episode of Futurama in which Prof. Farnsworth wistfully considers what might have been if he had invented what he calls the “Fing-Longer” (pic). Does having a fully developed idea of something and how to make it count as inventing that thing? And is this the kind of thing we would call an invention in the first place?