(Book 2 Chapters 7-13) How is Lucie the “last dream of [Sydney’s] soul?”

Lucie isn't so much a "last dream", as she is an inspiration. Caton's feeling for Lucie have inspired him to be a better man..... a better person.

I wish you to know that you have been the last dream of my soul. In my degradation I have not been so degraded but that the sight of you with your father, and of this home made such a home by you, has stirred old shadows that I thought had died out of me. Since I knew you, I have been troubled by a remorse that I thought would never reproach me again, and have heard whispers from old voices impelling me upward, that I thought were silent for ever. I have had unformed ideas of striving afresh, beginning anew, shaking off sloth and sensuality, and fighting out the abandoned fight. A dream, all a dream, that ends in nothing, and leaves the sleeper where he lay down, but I wish you to know that you inspired it."