One of the infant's middle names appears to pay tribute to Eugenie's grandfather, Prince Philip.
The baby - a ninth great-grandchild for Queen Elizabeth II - was born February 9 at London's Portland Hospital.
Sharing the latest photograph of the couple with their son, Eugenie wrote on Instagram: "Our hearts are full of love for this little human, words can't express".
"Thank you for so many wonderful messages".
The photos were taken by their "wonderful" midwife, the couple said.
Princess Eugenie leaving hospital with her newborn son Credit Karwai Tang
In the picture, the child is wrapped in blue while being held by his father and clutching his mother's finger.
She added that the images were taken "By our wonderful midwife".
A few days later Eugenie waved from the back seat of a 4×4 with her son beside her in a carry cot, while husband Jack drove them home from the hospital.
Until Baby Sussex arrives, the first 10 places remain unchanged - the Prince of Wales, the Duke of Cambridge, Prince George, Princess Charlotte, Prince Louis, the Duke of Sussex, Archie Mountbatten-Windsor, the Duke of York, Princess Beatrice and Princess Eugenie. He is a regular Master Brooksbank.
Eugenie married 35-year-old Brooksbank, a businessman, in October 2018 at St George's Chapel at Windsor Castle.