As posted above, rules about where to claim family benefits are all decided under EU Reg 1408/71. In this case, it willd depend on whether the partner of the OP is working in NI - if not, then CB paid in RoI, if partner is working in NI, then CB or equivalent is paid in NI. Early child supplement is also decided in the same manner.
However, the person will not lose out as, if the receive their benefit from the country with a 'lower' rate - then the other country will pay a 'top-up' to bring it in line with its own rate.
So, if OP has partner working in NI and child lives there, then NI will pay child benefit. However, RoI will then pay him a top up to ensure he is paid the same as if he were receiving full benefit in RoI.