Yeah, other than mischief, I'm unsure what Lady Jayne expected to accomplish by manipulating Lucy that way. Revenge on Mina for attracting Grayson's attention? Distracting Mina from Grayson?
She certainly wasn't trying to seduce Lucy (which would have been easy enough), and she can't really have expected that Mina would welcome Lucy's advances. Lady Jayne is obsessed with Grayson, not Mina or Lucy, so we have to assume that she was trying to disrupt the Grayson/Mina thing somehow . . . or perhaps trying to turn Lucy against Jonathan and Mina for some reason.