Hi Paul I have taken, copied and pasted below details from booklet 6, which outlines the criteria for identifying what is an eligible cousin.
Sonsorship by a relative
If you have a relative living in a ‘Designated Area’ who is willing to sponsor you, he or she will need to be
an Australian citizen, permanent resident or ‘eligible New Zealand citizen’ (see page 5). You or your
partner must be related to your sponsor as:
• a non-dependent child or non-dependent step-child;
• a parent or step-parent;
• a brother or sister, step-brother or step-sister;
• a niece or nephew, step-niece or step-nephew;
• an aunt or uncle, step-aunt or step-uncle;
• a first cousin; or
• a grandchild or step-grandchild.
If you are relying on sponsorship by a person related to your partner, your partner must also be included
in your application.
To ensure the processing of your application is not delayed, please include a diagram of your family
tree with your application. You will also need to include all relevant documents as evidence of your
relationship to your sponsor and their residential address.
If you are seeking an invitation for this visa on the basis of family sponsorship, when you submit your EOI
you will need to provide details of your sponsor. Your sponsor must be an eligible relative living in a
designated area.
Personally, I think it is irrelevant that your wife's cousin and her do not share the same paternal grandmother. You will notice above, step aunties, uncles, brothers, sisters and granchildren etc qualify as sponsoring relatives. Naturally, you wife and cousins parents' are step brother and or sisters. The issue is therefore: you cant be a a half cousin, there is no such thing, and a second cousin though not eligible would constitute the children of your sponsoring cousin. I hope this helps.