There are also similarities between The French Lieutenant's Woman and Martin Amis' novel London Fields, though on the surface they may seem strikingly different.
Their narrative styles are quite similar in that both stories are relayed to us by a narrator who frequently interjects the text to tell us his opinion of the action. Both narrators also consider themselves the authors of their respective stories. In this way both of the texts are postmodern ones, focused on the "self-conscious act of the author telling a story" (eNotes, The French Lieutenant's Woman: Style).
Secondly, both novels deliver a portrayal of English life and society, which might be pertinent to you in your search for an English novel. London Fields, however, is slightly satirical and conveys a much darker side to London at the turn of the millenium.
Each novel also has a sensual female character who inspires obsession in the men who become enamored by her. Plus, in the same way that Sarah remains an enigma throughout The French Lieutenant, Nicola Six in London Fields is a mystery to the reader. Is she the shy virgin she sometimes pretends to be with Guy? Or the cunning "murderee" she presents herself as to Keith?
Like thanatassa said, you must choose a novel that you can analyze alongside The French Lieutenant's Woman for similarities in style, theme, or whatever else it is you want to focus on in your essay.