Are PhD "minimum admission requirements" flexible? I want to apply to do my PhD at Melbourne University, where they require an honours degree with a "substantial research component equivalent to at least 25% of your final year."

I'm about to finish with a first class honours (all A+), but my only research project is equivalent to 1/8th of my final year. I also will do a summer research project this December though.

Will I still have a chance at admission even though I fail to meet their minimum requirement? As far as grades and courses go I believe I'm a good candidate for the actual research area.
Easiest solution, contact the University's graduate admissions office. Admission criteria will depend on funding source, if you or your supervisor have direct funding for the PhD admissions is usually more flexible, but if you're applying for a competitive scholarship you might struggle to compete against peers with MRes/AU Honours/publications. 

I'm assuming you're going through the UK honours system based on your question. The Aussie system is a bit different where and honours is more akin to an MRes in the UK (see here for more detail: https://en.wikipedia.org/wiki/Honours_degree).