My best guess is "His dream is to become a movie director," but I'm not sure what "になるという" means when put together like that. I know that "になる" means something like "becomes," and "という" means something like as/named/called, but I'm not sure how to parse it.
The basic sentence is this:
彼には 夢が あります
'He has a dream.' (polite)
This is a possessive argument structure1. In this argument structure, the possessor is marked by に, and the possessed is marked by が. Since in your example this clause is the main clause, the possessor is naturally expressed as a topic, so the underlying に becomes には, and we end up with the ～には～がある pattern.
Your example also has a noun-modifying clause (specifically, an appositive clause):
'a dream, to become a film director'
The noun-modifying clause is 映画監督になる (lit. 'become a film director'), and it is linked to the noun 夢 by the complementizer という. In this use, という is used grammatically to show the relationship between the noun-modifying clause and the noun 夢; it is not used literally to mean 'say', and that is why in this use it is usually not written in kanji as と言う.
The noun phrase 映画監督になるという夢 as a whole is marked as the second argument of あります with the particle が in your example:
彼には 【映画監督になるという夢】が あります
'His dream is to become a director.'
In this English translation, the possession is expressed with a possessive pronoun rather than a clausal possessive construction, and to become a director is related to dream with a subject–predicate relationship rather than an appositive relationship. It expresses the basic idea, but the grammar is very different!
If we wanted to make a slightly more direct translation, we could say something like:
'He has a dream, to become a director.'
But the other translation is probably a more natural way to express the same idea in English.
1 Japanese: Revised Edition, Shoichi Iwasaki, Chapter 6 'Argument Structures', p.107.
2 Japanese: Revised Edition, Shoichi Iwasaki, Chapter 9 'Noun phrase structures', p.202, p.210.