Mr Cain’s son James said it was a proud moment.
“Dad never sought recognition for his efforts, but as family, we are proud that this gesture has been made and we accept it on his behalf.”
As premier, Mr Cain was a driving force behind the creation of Melbourne Park, then known as the National Tennis Centre.
Since moving to the complex from Kooyong, the Australian Open has grown into one of the country’s biggest sporting events.
The renaming of the arena will occur in coming months and has been endorsed by Tennis Australia, which agreed to relinquish naming rights.
The announcement comes ahead of a state memorial service for the former premier that will be held at St Paul’s Cathedral at 11am on Monday.
Doors will open at 10am, with members of the public encouraged to attend.
Mr Cain’s family has asked that people consider making a donation to support the Victorian Bushfire Appeal in lieu of floral tributes.