The Queen will have the final say as it will be owned by the Crown. The prison at Wrexham Industrial Estate will house 2,000 inmates.
Work began last September on the Â£250m complex which is due to open in 2017.
A newsletter updating local residents on the construction said the name should embrace the history and culture of north Wales and the Welsh language.
Schools and community groups have been invited to come up with names for the new-build "super-prison" in Wrexham.