The house, which is a three-minute walk from University Hospital Southampton, was founded 32 years ago.
It provides accommodation for people whose relatives are being treated there and about one in five people who stay there are from the Bailiwick.
The charity hopes to sell 5,000 bricks and raise £500,000 through the campaign.
‘We think the house overall is going to be 22,000 bricks,’ said Amelia Firetto, Heartbeat’s fundraiser for Dorset and Guernsey.
‘We wanted a way to get individuals or corporates involved.
‘A build is such a big job, and we have supporters who have donated fantastic sums of money, but this seemed a nice way for an individual to still play a really significant part, and we can give them a bit of recognition back.’
Every sponsor will receive a commemorative Lego-style brick, and their names will go on a plaque in the house.
The work will bring the five bedrooms in the separate annex into the main house, creating a 24-bedroom building. It will also give residents better communal space, which the charity can use for events and meetings.
The rest of the house and the garden are being improved as well.
Ms Firetto said the project was on schedule.
‘Everything is so far going to plan, which is a bit of a miracle in the old building world, but we’ve had the weather on our side up until now,’ she said.
‘We will definitely be done in February, and then we’re hoping to hold the opening events more towards the spring so we can iron out any teething problems and finish the furnishings.
‘I’m not in the office all the time, so whenever I come in, it’s like the whole house is completely different.’
The house is staying open at reduced capacity while the work goes on.
n To sponsor a brick, visit heartbeat.co.uk/appeals/sponsor-a-brick-campaign.
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