In the hypothetical scenario that some ill befalls the monarch, Can Hand of the King call for a great council to elect the heir if the succession was not clear? For example, if Daenerys Targaryen passes away, can her hand Tyrion call a Grand Council to elect the heir (Presumably Jon Snow) and ratify his accession?
Yes the Hand of the King can assume the regency and title "Protector of the realm" in the interregnum and they can officially summon a Great Council of all the Lords of the realm to elect the new King or heir.
For historical precedence, we have Lord Brynden 'Bloodraven' Rivers, when King Maekar died unexpectedly in a battle without any clear heir, Lord Brynden assumed power as Hand of the King and called the Great Council of 233 AC in which King Maekar's youngest son Prince Aegon was elected.
So in the event that Daenerys dies without naming an heir, Tyrion would be well within his rights to assume regency and call a Great Council.