"The son, John Birnie, styled of Birnie, married Jean, daughter of James Hamilton of Broomhill, Bishop of Galloway, second son of Sir James Hamilton of Broomhill, baronet, a younger brother of Lord Belhaven, from whom the bishop seems to have acquired the lands of Broomhill.
The bishop had two sons, both of whom died without issue, and the estate of Broomhill, came into possession of his daughter Jean above mentioned, through whose right it devolved upon the Birnies. She was succeeded by her eldest son, John Birnie of Broomhill."
AND
"The estate of Broomhill, which is in the parish of Dalserf, Lanarkshire, remained in possession of the Birnies till about 1825, when, from the death of the last direct descendant, a lady, the estate was sold by her heirs to James Bruce, Esq., a native of the parish, who had returned from India, with a fortune."
https://www.electricscotland.com/history/nation/birnie.htmSo you have a John Birnie marrying a Jean Hamilton (doesn't that make them Hamiltons??), then the land that Jean Hamilton inherited was passed on to her son John Birnie.
1728 has John Birnie SNR passing land the John Birnie JNR
https://digital.nls.uk/histories-of-scottish-families/archive/95375127?mode=transcription"Upon the 6th of June 1717, the aboue John Birnie married for the second tyme his before spoken off cussen, Margaret, only daughter of Mr Alexander Miln, one of the ministers of Glasgow, and Ann Hamilton ((Hamilton!!!)), youngest daughter to the also before James, Bishop of Galloway."
Margaret Birnie (nee Miln) died in 23d March 1729. "She dyed at Edinburgh, and left of ishue James,* born at Edinburgh, 10th August 1720."
https://digital.nls.uk/histories-of-scottish-families/archive/95375175?mode=transcriptionJust 43 years later
"1773 The King has been pleased to grant his Royal Licence and Autho-
rity to James Birnie of Broomhill, Esq. in the County of Lanark, and his Issue, to take and use
the Name, and bear the Arms of Hamilton, the same being first duly exemplified according to the
Laws of Arms, and recorded in the Heralds' Office."
I would be wary of assuming anything about this family of Birnie-Hamiltons?
A sister of James or John Birnee of Broomhill could have married to a Brown(e) could have been the heiress and this could be where Brown(e) surname came from too.