The Visit Belfast office in Donegall Square North (30m from where we meet) has a left luggage facility which costs a few pounds normally.  Visit Belfast opens every day at 9am except Sundays when it opens at 11am. They can answer any questions on storage directly. 

Sunday storage

On Sundays an alternative is the Irish Tour Tickets office on Great Victoria Street, approx. 500m from city hall which also offers a luggage facility.  Price is £5 we think?