"meines" is the genitive case whereas "meinem" would be dative.
Generally you'll learn genitive by saying things like "Das Haus des Vaters" ("The house of the father" or "The father's house"). My usage above is a bit of an odd case where "In der Nähe" just happens to require a genitive object ("Haus"), while the preposition "Neben" just happens to require a dative object. I don't know if there's any way to tell what case is required in situations like these besides memorization.
Keep in mind though, that "neben dem Haus" is a prepositional phrase, and "in der Nähe" is a prepositional phrase, but "in der Nähe des Hauses" is a prepositional phrase followed by something else I'm not certain the grammatical name for.
It is a very odd thing to say. But it is what the German sentence means. You cannot change the sentence to something that sounds better to you, but means something quite different!
I am guessing that this sentence was included in order to teach that bei does not mean by.