I get to a Top30 PhD program in the US directly from New Zealand after an Honours year, so I believe I may contribute some inputs.
The Honours program at my university is the only one in NZ that is comparable to the Honours at the Group of Eight. But even so, I am placed at a significant disadvantage when applying for PhD in the US.
Here are my thoughts and I hope they could be useful for future applicants from Oceania; it would not be useful for those, say, in Canada or the US, as things are so much different down here.
1. I do have the anecdotal observation that fewer and fewer Honours students are directly admitted to a PhD program in the US and UK. Most of the Honours cohort would work in the industry after graduation.
2. RA opportunities in Honours year are very rare and the amount is decreasing every year. You need to reach out to many professors, and only a handful of which offer RA jobs. Perhaps a better way is to work as a full time RA after Honours graduation, and certainly in both Australia and New Zealand there are relatively more RA job opportunities outside university, although these opportunities are considered to be limited by the US standard as well.
3. Choose your Honours school wisely, because in essence you are choosing who will write your recommendation letters. Ideally the writers of your recommendation letters should come from top US schools, but even within the Group of Eight this is not guaranteed. Try to locate the best professor from your Honours school and do the Honours thesis with him/her.
4. In my view, the best Honours school in the region is UNSW. In fact, I think one student from UNSW get to MIT this year (from the Profiles and Results 2019 page, if I'm not mistaken). I'm not entirely surprised by that, because UNSW has the best professors in the region, they have excellent RA opportunities, and their Honours program has always been rigorous. Other than UNSW, I would recommend ANU and UniMelb (maybe Monash and UniSyd marginally), but I think they are not at the same level as UNSW.
Of course, this is only my subjective opinion, and is open to debate.
5. If unfortunately, you cannot secure a RA job because of limited availability and crappy quality, then perhaps the only way to get to a top PhD program is to try your best in Honours thesis, and become a mathematician.
There are many threads discussing the ideal preparation you need to get to a top PhD program. In general, a successful applicant should have mathematical preparation up to real analysis, with extensive RA experience and excellent thesis in economics. However, if you cannot have RA experience, then you need to do more advanced maths courses after real analysis, and of course get good grades. Such remedy might be able to get you to Top 30, but for sure not Top 10.
Without sufficient RA experience, if you only have an Honours thesis and study maths up to real analysis, you will be very lucky to be admitted to a Top 50 school.
But of course, not everyone could do advanced maths, and they are not as attractive as RA experiences in the eyes of top PhD schools...