First, you'll have to see what kind of agreements do you have with your employer as there is a good chance that the company would state they own the work for the personal project and it is their intellectual property. This can be important for some people as if someone finds that million dollar idea, they may want some additional compensation for that. Alternatively, some may want control over personal projects once they get past an initial prototype phase.
From Seven Ways to Boost Employee Morale:
Personal projects can provide an energizing break from regular
responsibilities and can serve as a source of innovation for a
company. Atlassian, a developer of collaboration software based in
Sydney, encourages creativity during its "FedEx Day." During this
event, all 62 employees can work on anything that excites them -- as
long as it is somewhat related to Atlassian products or processes, can
be completed in the allotted time, and is fun. Employees have from 2
p.m. on a Thursday until 4 p.m. Friday, giving them roughly 24 hours
to deliver a project (thus the name, FedEx Day). Then at a
presentation, participants show off the results of their projects.
From these ideas, Atlassian has adopted more than a dozen projects,
ranging from product upgrades to process improvements.
Another source would be How 3M Gave Everyone Days Off and Created an Innovation Dynamo to demonstrate that this isn't that new of an idea at least for one company:
3M launched the 15 percent program in 1948. If it seems radical now,
think of how it played as post-war America was suiting up and going to
the office, with rigid hierarchies and increasingly defined work and
home roles. But it was also a logical next step. All those early years
in the red taught 3M a key lesson: Innovate or die, an ethos the
company has carried dutifully into the 21st century.
Next, consider what kind of structure would be desired for your company? Is trying to get a deliverable product in 24 hours reasonable? Would it be better to let the people have a few months to get the project into a state to show to other employees? Would it be best for each person to be their own independent army in getting something done or could people work on personal projects together? There are several details to work out though my suggestion would be when you pitch this is to have several questions so that management can make some choices and feel like they have some ownership in this program rather than just rubber-stamping the idea.
In previous positions I could carve out some time for personal things that I could then pitch as ways to change process or add something new to how the company functioned. My current work as a Research Assistant does give me some latitude to find new project ideas and bounce them off of others.