Apparently, naming Regice “Regice” will not count as being a nickname, and therefore will not summon Regigigas.
Bulbapedia states that you need to fulfill certain criteria to summon Regigigas:
In Pokémon Omega Ruby and Alpha Sapphire, the player is able to encounter Regigigas in the Island Cave after capturing all three Legendary titans in their respective locations in that game. To do so, the player must go to Pacifidlog Town and speak to the girl in the northeast house. The player must also have a nicknamed Regice holding an ice-based item (a Casteliacone, a piece of Never-Melt Ice, an Icy Rock, an Icicle Plate, or a Snowball), as well as Regirock and Registeel in their party. Regigigas can be encountered starting from the day after catching the last of the Legendary titans, but cannot be encountered at night. As Regigigas is not in the Hoenn Pokédex, it can only be battled after capturing/defeating GroudonOR/KyogreAS.
I tested this by capturing the three Regis yesterday, and trying to summon Regigigas today (daytime). When I tested with the three Regis in my party, already having met with the girl who gave me the correct information, Regice holding a Never-Melt Ice, and Regice being nicknamed “Regice”, the following message appeared:
"...?"
"There seemed to be some presence here, but it disappeared in an instant...”
This leads me to believe that since I fulfilled all criteria but the nickname, the nickname is what prevented Regigigas from being summoned.
Right after, I headed to the Name Rater to change Regice to “Regiice”, and I headed back to the Island Cave. When I entered, I was greeted with the following text:
"The very earth is quaking! Is something approaching?!"
"Zut zutt!"
Regigigas then appeared, and I was able to battle it.
Hope I helped!
