Note that the ESTA is not a visa, it is merely an authorisation to travel to the USA by air or sea and request entry under the Visa Waiver Program (VWP)
He can't apply for an extension of status because he entered on the VWP.
He could try leaving the USA, coming back and requesting another entry under the VWP but he runs the risk of being denied entry. There is no hard and fast rule on time in VS time out but the VWP is "intended for short occasional visits". If you do use this tactic it is probablly advisable to go to somewhere other than the countries that have special rules regarding re-entry (Canada, Mexico, some islands).
He could leave the USA and apply for a visa but visas take time and again there is no guarantee he will get it.
All in all if you want to spend more time together your best option is probablly to take a holiday together somewhere other than the USA.