Brady is born on Parliament Street, Dublin, the second son of Francis Tempest Brady of Booterstown, a manufacturer of gold and silver thread, and his wife Charlotte Hodgson, daughter of William Hodgson of Castledawson, County Londonderry. He is baptised at St. Werburgh’s Church, Dublin. He is the brother of Sir Nicholas Brady, Lord Mayor of Dublin, and uncle of the eminent ecclesiastical historian William Maziere Brady.
The Bradys are an old and distinguished Munster family who are particularly associated with the town of Bandon, County Cork. Probably the most celebrated of his ancestors is the poet and psalmist Nicholas Brady (1659–1726), who collaborated with Nahum Tate, the Poet Laureate, on New Version of the Psalms of David.
Other notable forebears include Hugh Brady, the first Protestant Bishop of Meath (d. 1584), his father-in-law Robert Weston who, like Maziere serves as Lord Chancellor of Ireland, and the judge and author Luke Gernon (d. 1672), who is now best remembered for his work A Discourse of Ireland (1620), which gives a detailed and (from the English colonial point of view) not unsympathetic picture of the state of Ireland in 1620.
In politics Brady is a Liberal and supports Catholic emancipation. He sits on a commission of inquiry into Irish municipal corporations in 1833. He is appointed Solicitor-General for Ireland in 1837 and Attorney-General for Ireland the following year. In 1840 he is appointed Lord Chief Baron of the Irish Exchequer. In 1846 he is appointed Lord Chancellor of Ireland and serves in that office, with short intervals, for the next 20 years. He retires in 1866 and is made a baronet, of Hazelbrook in the County of Dublin, in 1869. His appointment ends the practice which grew after the Acts of Union 1800 of appointing only English lawyers as Lord Chancellor of Ireland. He sits on the Government Commission on Trinity College Dublin in 1851, and is nominated as Vice-Chancellor of Queen’s University Belfast in 1850. All through his life he shows a keen interest in education.
According to Elrington Ball, Brady’s Lord Chancellorship is notable for its length but for nothing else. Ball calls him “a good Chief Baron spoiled to make a bad Chancellor.” By general agreement he had been an excellent Chief Baron of the Exchequer, having a reputation for being fair-minded, courteous and approachable, but in Ball’s view the more onerous (and partly political) office of Lord Chancellor is beyond his capacity. Unlike some judges whose training had been in the common law, he never quite masters the separate code of equity. Delaney takes a somewhat more favourable view of Brady as Lord Chancellor, arguing that while his judgements do not show any great depth of learning they do show an ability to identify the central issue of any case and to apply the correct legal principle to it.
An anonymous pamphlet from 1850, which is highly critical of the Irish judiciary in general, describes Brady as being unable to keep order in his Court, and easily intimidated by counsel, especially by that formidable trio of future judges, Jonathan Christian, Francis Alexander FitzGerald, and Abraham Brewster. The author paints an unflattering picture of Brady as sitting “baffled and bewildered” in a Court where he is “a judge but not an authority.” On the other hand, Jonathan Christian, who had often clashed with Brady in Court, later praises him as “no ordinary man” despite his shortcomings as a judge. He describes him as “independent-minded, patriotic, natural and unaffected.”
Brady is a founder member of the Stephen’s Green Club and a member of the Royal Dublin Society and the Royal Irish Academy. In addition to the arts he shows a keen interest in science, especially after his retirement. Like most judges of the time he has both a town house in central Dublin and a place some distance from the city centre. His country house is Hazelbrook, Terenure, Dublin. He changes his town house several times, settling finally in Pembroke Street.
Brady marries firstly Elizabeth Anne Buchanan, daughter of Bever Buchanan, apothecary of Dublin, and his wife Eleanor Hodgson, in 1823 and they have five children. Elizabeth dies in 1858. In 1860, Brady marries Mary Hatchell, daughter of John Hatchell, Attorney-General for Ireland and Elizabeth Waddy, who survives him. He dies at his house in Pembroke Street on April 13, 1871. He is buried in Mount Jerome Cemetery.