Andrew Bridge's grandmother Valerie said: "We are delighted. It is at least something and that is all we were asking for, all we wanted was another search.
"It might not come to anything but people want them to do it and they are trying. It seemed too quick, just two days and we were saying 'if only they could do it (search) for a bit longer'.
"You never know what could happen."
The families of the sailors have travelled to London to meet foreign minister Hugh Robertson.
Leading figures from the yachting world are among more than 175,000 people who have signed an online petition supporting the campaign to resume the search.
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Wednesday afternoon's weather for the east of the region
Wednesday afternoon's weather for the west of the region
A talented young sailor who represented Great Britain hanged herself in Hampshire the day after her 17th birthday, an inquest has heard.