the closest name that I can see is O'Hea - and the vast majority of these are located in Co. Cork, nearly 60 with 1 in Co. Armagh
Im using Griffiths valuation entries to get a rough count - see the free site at :
http://griffiths.askaboutireland.ie/gv4/gv_family_search_form.phpO'Hay gives no results anywhere in Ireland and O'Hey returns just one entry in Co. Limerick, it may also be worth trying other variations such as O'Hare ..
(Griffiths is a land valuation survey carried out in the mid 19 century)
without some additional clue it will be very difficult to start a meaningful search for Lawrence. Do you have any other clues ?
e.g. father's name or occupation, a family trade , a vague place name - mountain, river, landlord or estate etc... sometimes a little clue can help narrow things down.
Shane