A word-for-word translation in English could be: “from the vicinity of Swann’s place”. That’s not a perfect translation: the range of du côté de is broader than the vicinity, it also includes the place itself. “From a small area centered on Swann’s place” is closer to the right meaning.
Coming on to your question proper, the preposition chez admits an unusual construction (in addition to normal preposition-like use, as in “Je vais chez Swann”). In this construction, chez behaves somewhat like an article, but it is an article that stacks: it must be followed by a complete noun group which may itself include an article: “chez Swann”, “chez lui”, “chez le boulanger”, “chez mon cousin” are all valid noun groups. This noun group may in turn be preceded by a preposition.
In “du côté de chez Swann”, the noun group “chez Swann” is a complement of the noun “côté” introduced by the preposition “de”.
In the Trésor de la langue française, this construction is described by the second bullet point under A.1.a:
En raison du sens « dans la maison de », le groupe prépositionnel peut être lui-même précédé d’une autre préposition à valeur locale, le concept « dans » étant alors neutralisé et chez signifiant « la maison où habite…, séjourne habituellement… »
“Du côté de” is not a preposition, but it is a group of words acting like a preposition. Note that chez can only be used in this way when it is in a context that invites a location. For example, “I’m coming from his home” can be translated as “Je viens de chez lui”, but “I visited his home” cannot be translated as “*
J’ai visité chez lui”, you need to translate the concept of “home” explicitly, e.g. by “J’ai visité sa maison”. (The idiomatic translation for “I visited him” would be “Je lui ai rendu visite”, “J’ai visité sa maison” implies exploring the house itself.)
I think you want to match the French and English constructions to closely there. “Chez Pierre” has many meanings beyond “at Pierre’s”.
Je suis chez Julie.
I’m at Julie’s.
Je vais chez François.
I’m going to François’s.
Je viens de chez Pierre.
I’m coming from Pierre’s.
J’habite à côté de chez Germaine.
I live near Germaine. (literally “next to/near Germaine’s place”)
Il m’a fait visiter chez lui.
He showed me around his place. (literally “He had me visit his place.”)
Je rentre chez moi.
I’m going home.
There are others, more remote uses:
Tout me plaît chez elle.
”Everything about her pleases me”
La couvaison est une tâche partagé chez l’hirondelle des rivages.
Sand martins share turns brooding. (literally “Brooding is a shared task for sand martins”)
Du côté de and chez need not be taken at face value as if it were:
Du côté de la ferme, chez Grosjean, il y a une étable.
These terms are very general, have abstract connotations, and the novel is a psychological study. I suggest the explicit but heavy:
Insights into Swann’s acumen / mood / mind
“Swann’s way” is much smarter, of course.
Swann’s Way is a pretty clever translation and close to the same idea. But it also carries some interesting connotations if you think about all the usages of way.
Examples: “Hey, are you going Grandma’s way? If so, take her these eggs.”
American English: “I was out California way.”
So as du côté de chez Swann refers to a place or area, so to can way in English.
That being said it is not the most common usage, and so Swann’s Way brings to the mind things like Swann’s path or Swann’s road and most interesting to me is Swann’s manner all of which enriches this translation because all of the extra English connotations flavor the idea of the book expressed by the title.