My son signed a 12 month lease that started on the 15th of June. He paid the prorate. Does his lease begin the 1st of July? When does it end?

... Does his lease begin the 1st of July? ...

What does it say? The language of the lease controls. Only if it is unclear or ambiguous, do you have to look further.

For example, you say it started on 6/15. If the language of the lease says it starts on 7/1 but he was allowed to move in early, you would be correct. If, on the other hand, the lease says that it begins on 6/15, that would be when it begins.

... When does it end?

365 days from when it begins.