Hello, first off: no there is no specific name for Regice to nickname it as far as I know... I hear ya, I don't like to nickname my Pokemon either, but what I did was just nickname my Regice, and then later "change it back" to the name Regice at the Name Rater :) As long as you're the OT you can rename him as often as you like so it doesn't matter really... You can just give any nickname to Regice (let's say Icecream), and after you caught Regigigas, you return to the namerater to change the name back to Regice :) To be safe, I would just pick another name than 'Regice' though.
Also, you need all the Regi's in your party in order to catch Regigigas and wait a day after you caught the last 'golem' Pokemon in order for him to show up at Route 105. Also, like you mentioned Regice needs a nickname AND an ice-related item such as Icy Rock (Weather Institute), Nevermeltice (Shoal Cave) or a Castelia Cone (Trick House).
Well, I hope you have found an answer and good luck! :)