Proof that it satisfies what the customer needs or wants - something which can't be demonstrated until the would-be solution reaches some level of demonstrable maturity.
For many engineering solutions, you can talk about them earlier, handwave, present, but nothing conveys that it actually works, or sometimes even makes sense, short of a working demonstration.
Why so quick to assume it's a "pet" project? If the developer is doing it right, they are making exactly what the customer needs, but the customer does not have the developer's expertise to recognise what will work at an early stage.
We are not talking about things everyone understands and could have an opinion on, like say the colour and layout of the login window.
Take something that requires deeper work and hired expertise. For example suppose you've started to design a new kind of GPU technique that will double the customer's calculation throughput for the same power consumption. They hired you for your expertise to solve that sort of problem. They have no idea how it works though, and for the first month you have only an outline of a new algorithm, that only peers at your own level could understand well - and you don't have any peers in that area. And most of figuring out the details is on paper and whiteboards, math and logic along with throwaway experiments to measure. The first line of code in a working demo is not possible to figure out until 3 weeks in, and it takes only 1 day of coding and 2 days of polish after that to finish.
That sort of thing isn't a pet project. It might not even be fun; it might be quite stressful, after all it's a higher risk to the developer to pursue such paths. It doesn't fit into daily PRs and isn't even a coding problem at first. Yet it's exactly what the customer needs, what they said they hired you to accomplish, and what they will appreciate if it has reached the point of "proof" when shown.
It will be incorrectly assumed to be the developer's pet project if it's demonstrated before it works, though. Thus the reason to build it up to "proof" level, if you're sure the customer needs it.