I'm pretty sure that currently the club will only authorise share transfers in very limited circumstances, e.g. legacies
However HSL membership provides an indirect holding. HSL members collectively hold just over 15% of HFC shares.
As others have said above the only way to become a 'part owner' of Hibs at the moment is through membership of Hibernian Supporters.
You can sign up and make payment or part payment towards the £225 membership fee on behalf of someone else here and once full membership is achieved your son would become a part-owner of the 15.4% stake in the club held by Hibernian Supporters.