Mary Shelley would likely give Han Solo a more tragic backstory. She was good at creating characters with a sense of melancholy. Han Solo's orphan status could be expanded upon, and his relationships, like with Chewbacca and Qi'ra, could be written with a deeper emotional resonance. His actions would be more driven by his past traumas and his search for meaning in the vastness of the galaxy, much like the characters in Shelley's works are often driven by their own inner demons.
I think Mary Shelley would emphasize Han Solo's relationship with nature, or in this case, the space around him. In her works, nature often played a role in the characters' development. Han Solo's experiences flying through space, dealing with different planets' environments could be described in a more poetic and philosophical way. His character would not just be a swashbuckling hero but also someone who is in constant dialogue with the universe around him, much like the characters in Shelley's stories are often in tune with their surroundings.
She might write Han Solo as a complex and brooding character. Given her inclination towards exploring the human psyche, Han Solo could be portrayed with more internal conflict. For example, his decisions to be a smuggler and his later involvement in the Rebellion could be explored in a more psychological depth, similar to how she dissected the motives of her characters in her novels.