I've never heard of anything about aspen trees over here. There's a river in Cumbria called the Waver from OE/AS waefre meaning 'winding'. The -ly ending means 'meadow' or more precisely 'clearing'. There's a town called Waverton in Cumbria ('farmstead by the river Waver') and another in Cheshire ('farmstead by the swaying tree')...see how the root of 'winding' is manifesting here?
It seems as if someone has decided that 'winding' leading to a 'swaying' tree has given license to assume the swaying tree was a Quaking Aspen. Involves a certain leap of logic and without more proof I wouldn't go there.
Waverly would most likely mean 'meadow by the river Waver' or if you like 'meadow by the winding river'. If one could link it to regions out of Cumbria then the 'meadow by the swaying tree' would be an alternative meaning. As a placename, it's been folded in and as a surname it's rare.