The street is named after the philanthropist Thomas Pleasants (1729-1818) who donated money towards the establishment of the nearby Meath Hospital. Pleasants was born in County Carlow in 1729, son of William Pleasants and his wife Grace Edwards. 

His grandfather was Thomas Pleasants, alderman of Dublin. His grandfather had leased a large piece of land near Capel St. from Dublin Corporation, which Pleasants inherited some time after his grandfather's death in 1729. 

His initial income derived from this property, though he also had dealings with his cousins the Pasleys, who were wine merchants at 9 Abbey St. He married in 1787 Mildred Daunt, second daughter of George Daunt, surgeon in Mercer's Hospital. He and his wife (died 1814) were buried in the churchyard of St. Bride's Church. 

Among his donations were over £12,000 in 1814 for the erection of a large stove-house near Cork St. for poor weavers in the Liberties, £8,000 for the building of the Meath Hospital, and his own house (67 Camden St.) for the provision of a girls' school and orphanage, along with £1,200 per annum to run it and extra money provided for dowries for the girls (only applied to Protestants, though), to do this Pleasant's Asylum for Female Orphans on Camden st. was founded in 1818 it closed in 1949. He donated his large library and a large sum of money to the Royal Dublin Society.

 Press Up Hospitality Group is Ireland’s leading leisure and hospitality company and they are located on Pleasants Street. Press Up Entertainment is a cinema, hotel, pub, retail and restaurant operator based in Dublin, Ireland. Ownership is shared between Paddy McKillen, Jr., son of Paddy McKillen, and Matt Ryan. Their properties include the Clarence Hotel, the Dean Hotel, the Dean Hotel Cork, the Devlin Hotel, the Mayson Hotel, the Irish franchise for Wagamama and the Stella Theatre cinema chain. They also operate two formerly franchised Tower Records locations, which have outlasted the US-based parent chain by many years. Most of their new properties are located in new developments by Oakmount, a property developer with the same owners but operated independently. They have announced an entry in to operating bowling alleys, with a development at Dundrum Town Centre.