Hayley Gascoigne, who was in her 30s and from Scunthorpe, Lincolnshire, was taken ill in the Combined Court Centre in Hull on Thursday.
Witnesses said court staff and paramedics tried to revive Ms Gascoigne, but she was taken to hospital where she was pronounced dead.
A post-mortem examination is due to be carried out.
More on this and other stories from BBC East Yorkshire and Lincolnshire live
Her family said in a statement: "As a family we are trying to deal with the loss of Hayley. She will be dearly missed by all.
"Her death came as a terrible and sudden shock and we would politely ask that our privacy be respected at this very difficult time."
A Humberside Police spokesman said: "A woman in her 30s collapsed in the public concourse area of Hull Crown Court at approximately 3pm.
"She was taken to Hull Royal Infirmary, where she was sadly pronounced dead."
Ms Gascoigne's friends posted Facebook tributes to her.
One said: "Lost for words. Absolutely heartbreaking. Thinking of your four beautiful children. Rest in peace Hayley xx".
Another said: "She was a lovely, warm kind person and this is just so tragic. Love n strength to all her family n friends."
A mother-of-four died after she collapsed in the concourse of a court building.