Dame Siobhan Keegan took office today as Northern Ireland’s chief judge and there has been some debate on Twitter over her decision to style herself ‘Lady Chief Justice’ rather than ‘Lord Chief Justice.’
‘Lord Chief Justice’ is the statutory form of the title, and some have argued that she can’t change it unless the relevant legislation is amended. Others (such as Conor McCormick) have argued that sections 36 and 37 of the Interpretation (Northern Ireland) Act 1954 render a statutory change unnecessary.
I’m agnostic on the question, but the case of Dame Elizabeth Butler-Sloss may be illuminating. When she joined the Court of Appeal of England and Wales in 1988, she was styled ‘Lord Justice Butler-Sloss’ since that was the title used in the Senior Courts Act 1981 (known as the Supreme Court Act 1981 until 2005). In 1994, the Master of the Rolls issued a Practice Note allowing her to be styled as ‘My Lady, Lady Justice Butler-Sloss’ in court. However, the Practice Note stated that the masculine version of the title would remain the official one until Parliament amended the Senior Courts Act 1981. This change was eventually made by section 63 of the Courts Act 2003. It has been argued that section 6(a) of the Interpretation Act 1978 rendered the statutory change unnecessary, yet for whatever reason the Master of the Rolls in 1994 and the drafters of the Courts Act 2003 evidently felt that a statutory change was desirable.
As a practical matter, the question is purely academic. Even if Dame Siobhan’s title is legally ‘Lord Chief Justice,’ the Butler-Sloss precedent shows that day-to-day usage can diverge from the official usage.
A Venerable Puzzle 
I am not sure that the comparison to Lady Justice Butler-Sloss is entirely valid. “Mrs Justice” and “Lady Justice” are personal styles rather than the title of a singular office.
Brenda Hale was famously the only female recipient of a life peerage under the Appellate Jurisdiction Act 1876. I can’t find the text of her letters patent nor footage of her introduction, but the Gazette reference describes her as being a Lord Justice of Appeal and becoming a Lord of Appeal in Ordinary, though of course her title is Baroness not Baron. In at least the last two years it has been the norm for the Gazette to say “Lady Justice” when mentioning female appointments to the Court of Appeal, but notices as late as 2005 still said “Lord Justice”.
Notably there have been five women to hold the office of Lord Keeper of the Privy Seal, six to be Lord President of the Council and one to be Lord Chancellor. I don’t know how many women have served as Lord Lieutenants of various counties.
A confusing case occurs when looking at the ushers and heralds of certain orders of chivalry – it seems that a woman appointed as gentleman usher becomes Lady Usher (Sarah Clarke for the Black Rod, Dame DeAnne Julius for the Blue, Dame Amelia Fawcett for the Purple) but a woman appointed as king of arms does not become a Queen of Arms (Lady Ashton of Upholland for St Michael & St George).