Er wechselte das Öl des Autos des Sohnes meines Bruders.
Does it make sense?
In short: yes, that makes perfect sense. The "son of my brother" could also be called "my nephew" ("mein Neffe"), but grammatically it is OK.
If you are asking from an english POV: one of the differences between English and German is that German has a full-blown case system, which English lacks. Even the English Genitiv (e.g. "my fathers house) is not a real (full-blown) case any more, but a "Fallrest" - and one of the places where that shows is when building chains of objects. One can say:
my father's car
but
my father's car's tires
would be wrong (or, as @RDBury pointed out in a comment, at the very least unidiomatic). Instead, constructions with "of" are used for these occasions, i.e.
the tires of my father's car
the tires of the car of my father