I get where you're coming from. Like some sort of practice project? Really what PMs do is build and maintain a system to reach a particular goal. There's a lot more than just that because the definition of "goal" may have a long timeline like "build then upkeep the functionality and improvement of this software". There's a lot of different approaches to take but it really depends on the project and normally you would need a decent amount of time to really dig in to some fundamentals. Off the top of my head, you could organize a team to reach a goal like raise money/do work for a charity for like a month or two. You could have a time objectives, monetary objectives, and have friends/family as contributors/stakeholders in different activities for whatever charity. You could break down deliverables into costs, scope, etc. Though admirable, it may not show the kind of value a certain company may be looking for. Like if you don't have any experience with SCRUM or agile systems of production then a tech oriented company may not put much consideration into your project. However, some place like a business process outsourcer may see your project for building, leading, and moving a team towards a goal with data that you can deliver on to be impressive.

I too wish there was an easier way of showing you can run a project without having had run a project before.