This is a very interesting idea. Have you, or anyone else on HN, ever seen this in action? I would LOVE to hear stories about this, either successes or failures.
It's called 20% time and it works OK. I always used it when I was at Google, although most people didn't.
20% is not specifically for "improving the organisation" of course. It's most famous on the outside for the new products it led to, like Gmail. But actually most 20% projects were small internal things intended to smooth the rough edges off a particular tool or process, or to improve the company in some way. If the particular bee in your bonnet was a type of bug that cropped up frequently in other people's software, making a linter to spot it and driving adoption through the organisation would be a good 20% project, for instance.
20% is not specifically for "improving the organisation" of course. It's most famous on the outside for the new products it led to, like Gmail. But actually most 20% projects were small internal things intended to smooth the rough edges off a particular tool or process, or to improve the company in some way. If the particular bee in your bonnet was a type of bug that cropped up frequently in other people's software, making a linter to spot it and driving adoption through the organisation would be a good 20% project, for instance.