Mornington Peninsula is great but New Year's period will be very, very crowded.
Great Ocean Road is very crowded at that time of the year too, accommodation won't be cheap either.
Plenty of beaches and things to enjoy for your kids. I'd avoid the crowded ones at Sorrento, Portsea, Rye and Dromana though. Westernport (eastern side of the Peninsula) is quieter.