Finch-Fletchley would be her married name when she married Mr Finch-Fletchley and, unless she is like Sirius Black's parents who both had the same surname prior to marriage, her maiden name was something different.
It is not how it is in North America. In the UK, hyphenated names are a sign that the person is a blue blood. Somewhere down the line, a Mr Finch married a Ms Fletchley, but the Fletchley family was of a higher status than the Finch family so the children were Finch-Fletchleys (and their children and the children after that) so as to distinguish that line from other Fletchley lines.
Look up Winston Churchhill on the wik since his true surname is Spencer-Churchill. Camilla Parker-Bowles is another example where Parker-Bowles was the surname of Camilla's first husband which she married when Prince Charles dumped her, due to family pressure, because she was Catholic and the King is supposed to be the head of the Anglican church. (Vaudree (talk) 01:54, January 6, 2016 (UTC))