A portfolio of maybe simple (but well-thought out) products built out. A lot of the job is about championing the product vision and distilling it down to requirements, as well as tying the loose ends (how well does the product work in foreign languages? on different platforms? for blind users? etc.), so attention to details is valuable.

All very good points, to which I'd add:

A product manager needs to have the ability to communicate effectively with people across a diverse set of roles, including customers, marketing people, designers, developers and upper management - and they probably all have different points of view on what's important to have in the product.

Thanks for the great advice. I have just been struggling to decide between different projects. Do you think a consumer mobile app would have enough degree of difficulty to impress companies?

Yeah, something that you've built that when it comes up in the interview, you're able to explain the goals you were trying to achieve, the trade-offs and design decisions, perhaps some A/B tests you've run and metrics you tracked.

Even if the app is simple and has a grand total of 2 users, it should provide decent conversation fodder and peg you as disciplined, detail-oriented and big-picture-focused. For example, if you have not built something obvious in (internationalization, in-app purchases, deep links), throw in an explanation of how you're aware of it and made sure it fits into the next release.

One quote from Sam Altman' startup class that stuck with me was (I believe it was from Brian Chesky) someone asking two builders what they were working on laying bricks. One said "I'm building this wall", another said "I'm building the cathedral". Be the second guy.

I don't think it is really about the difficulty of the project. Like prostoalex said above, it is more about the product being well thought out- ease of use, design, UX, its ability to solve the problem it is trying to solve, etc. Also, beyond the product- communication is key (as greenyoda explained above), so is the ability to prioritize problems that you want to solve (at its core, building products is about providing solutions to problems) and managing stakeholders' often competing interests and expectations.

That last part about managing stakeholders' competing interests and expectations might be the most important part. As an engineer, I have been furious with bad product management pushing near impossible designs and setting ridiculous deadlines without consultation in the past.

If I had to hire for a product manager, an indication of the skills associated with this would be what I would look for. Critical thinking and diplomacy are crucial skills, in addition to all the things you have mentioned.

Do your research and show how you can contribute. If you know what my products are, who my users are, and have insights for how to improve the products to better meet their needs I'd give you bonus points in the interview.

Working with teams is also important. I'd definitely look for a portfolio like the other commenters suggested, but also that you worked with others on some of these projects. Going from hacking something out yourself to managing a product built with other busy people is a big step.