The surname 'Monaghan', with or without the 'g', is the English version of a Gaelic name O Muimhneachain (pronounced ' O Mweenacawn') which means 'descendant of a Munster man'. Munster is the southern province of Ireland, including countiers Cork, Kerry, Tipperary etc. The more common surname Moynihan has the same origin.
The surname has no connection with the place - they just both happen to have the same spelling. There are only a few families in Monaghan called Monahan - they have dropped the 'g' probably to avoid confusion.