I am currently researching the family of Garnham and I have:
William Garnham (c.1602-1669) m.1629 Mary, dau. of Rev. Petley Wyborne, Rector of Elmstone. William Garnham was of Farnborough and/or Catmore, both in Berkshire and was Serjeant of the Countinghouse to his Majesty (Charles II). They had 2 sons and 5 daughters, one of whom was Frances who married Bernard Edwards and they had a son called Garnham Edwards, who had a son called Garnham Edwards, who had a son called Garnham Edwards.
Now I think that Frances' sister Elizabeth married Robert Barnes and had Joseph and Mary (both living 1710 according to the will of Petley Garnham, brother to both Frances and Elizabeth).
It is therefore not inconceivable that your Garnham Edwards Barnes derives from these familes.
HTH,
Caspar