Hi Denise,
I would like to give you my little, little help... don't know if it can be useful, but in the doubt....
I read all the 8 pages of your great-grandfather's naturalization papers on the site of the National Archives of Australia:
http://www.naa.gov.au/collection/recordsearch.aspx
on page number 7 Peter signed the documents as Peter
Bignone
on page number 6 Peter Begnone declared to be born on January 25, 1864 in " Grozu" province of Lombardy in the country of Italy...
As PeterTiber wrote, Grozn doesn't exist in Italy, neither Grozu but I found that there is a little Comune in Lombardia Region called
Grosio (4.820 inhabitants, Sondrio Province) ....
Moreover, the italian pronunciation of the "s" of Grosio is almost like the "z" pronunciation...
Don't want to create illusion but, why don't ask to this little comune if they have Civil records for the year of birth of Peter?
From this useful site:
http://italia.indettaglio.it/eng/cognomi/cognomi.html
you can also see that the Bignone actually in Lombardy region are only a few people
Number of Bignone: 5.28 Cologno Monzese - Milano
Number of Bignone: 4.74 Varese - Varese
Number of Bignone: 2.62 Arzago d`Adda - Bergamo
Number of Bignone: 2.29 Milano - Milano
Maybe writing to these persons can help too....
From the italian whitepages:
http://www.paginebianche.it/index.html
the Bignone actually listed are 191... mostly of them in Liguria region... but as in 1864 Lombardia region was very well defined I think that very likely your great-grandfather was just from Lombardia...
This is a map of Italy during the year 1861:
http://users.dickinson.edu/~rhyne/232/Five/Map1861.html
I hope I didn't made confusion ...
Laura