Blimey, thats a bit of a pickle. Sorry, but it sounds like they probably have the right to ask for the extra money - would you want to put a relationship with your relative at risk for 300 quid?
Im sure someone with law knowledge will help, but didnt mr tigwig realise he was doing 1/2 the work they had asked him to do?
if you have accounts then of course they will be able to prove what they paid.
in my mind, if mr tigwig quoted 12 punds per sq meter then thats what he should get paid, unless he is charging by the day, time taken to do it dosent count - he could have sped up and taken 3 days, or slowed down and taken 12 days, but he will still have done the same amount of paving