The sentence is correct.
The version with a possessive, his being..., is employed when the -ing clause acts as a 'gerund', in the same contexts where a noun phrase may occur. In your second example, for instance, the clause my being late every day is the object of the preposition to; you might equivalently say "He turns a blind eye to my tardiness." The possessive form his acts as a determiner and marks the -ing form as 'nouny'.
In your first sentence, however, the -ing clause is an 'absolute', a syntactic supplement which is not integrated into the syntax of the main clause. The verb BE is cast in the -ing form to mark the clause as subordinate to the main clause, but in other respects the -ing clause is structured like an independent clause: its subject takes subject form.
You should note, by the way, that both the possessive and the subject forms alternate with an object form:
He turns a blind eye to me being late every day.
Him being the eldest son, his father expects him...
With gerund clauses, the object form is common in colloquial use, but traditionally deprecated in formal use. Absolutes are quite formal and are so rare in colloquial use that they tend to sound odd with any form of pronominal subject. I recommend that in your own writing you restrict yourself to nominal subjects, where the object/subject contrast does not exist.
Rob being the eldest son, his father expects him...