1) Define success before doing anything. Pick specific companies or a specialty you want to end up with first.
2a) Take the three months (or however long you can afford to invest) and work backwards. Figure out the most impressive and relevant project that can be reasonably be accomplished in that time. Make sure it's part of a hot trend because the time spent is probably the same regardless.
2b) In parallel, network and build the best quality contacts you can in your target area. Blog about any interesting observations made as the project progresses. Come right out and say in your posts that you are doing this in hopes of building skills, experience, and proof of your abilities, and say where you want to end up.
3) When its done, make it public online make sure it presents with visual appeal and with cogent explanations. Ask for feedback from your contacts, because that's your excuse to show off: "sure was challenging but I was able to go from zero to learning and building this in three months!". Ask for feedback because it's your chance to ask about openings and interviews. Ask for feedback because you really will need it.
How does somebody new make a good project choice, when taking into account everything that would actually be impressive and help land a job would require years of experience to know? You can't without feedback or validation, so don't make the mistake of choosing solo. Just ask people who are in the target area to help you decide. Lots of people would be willing to give input and it's more chances for networking and follow up conversations.