Garthwaite surname concentrations appear to suggest the name originated in the Darlington area of North Yorkshire in Northern England.
The words Garth and Thwaite are Viking in origin.
The Viking Network site has the following information:
Yorkshire Dialect Word: garth
Generally accepted meaning: Small grass enclosure adjacent to a house.
Old Norse source word: garðr
Comments: Obviously related to the Standard English garden, French jardin, German garten, etc.
Yorkshire Dialect Word: thwait(e)
Generally accepted meaning: Village or small settlement
Old Norse source word: tveit
Comments: Now found only as an element in place names (e.g., Linthwaite, Micklethwaite, etc) and as a family surname.
Both these terms have pretty general meanings and there are a great number of 'thwaite' suffixed place names across the North of England. My interpretation of the above is that Garthwaite means Garden Village.
I checked the The Oxford Names Companion (Published by Oxford University Press ISBN 0198605617) for a definition, but it only listed Garth and Thwaite as separate names.
I have found a definition from the Dictionary of American Family Names (Published by Oxford University Press ISBN 0195081374):
English: habitational name from a lost place in northern England; the second element of the place name is probably Old Norse þveit 'clearing'.