One suggestion from Madeleine Flannagan :
Have her/him do the Tertiary Foundation Certificate (1 year, costs a total of about $540) the year before she wants to start her degree.
Pass this and you have as much shot as anyone with good NCEA or Cambridge results at any Auckland Uni degree program – even restricted entry ones. Of course what helps with restricted entry courses are A’s so tell her to pass it well 
The TFC course is an excellent intro to Uni – it is more hands on so there is a lot of guidance, here is how to write essays to Uni standard, it is on campus, run by Uni lecturers, so it is way better than NCEA or Cambridge.
My homeschooled daughter was accepted to Auckland Uni’s TFC program one month after her 16th birthday on the strength of her diagnostic test scores (they run this test twice a year – it is free to sit).
The other way to get into Uni without NCEA or Cambridge is to sit a few no-requisite, easy entry papers at Uni, pass them well, then simply point to them on your application.
The purpose of tight entry criteria is to weed out applicants who do not have what it takes to pass at Uni – Uni does not care so much about NCEA or Cambridge, they care about whether or not you can do the course, handle the workload so TFC and passes in other Uni papers are fine because they answer that question.
 They have a free trial if you want to check it out. 
