I think I first heard (or perhaps I remember it because it was one of the classic lines in anime) となる in Death Note:
Boku wa shinsekai no kami to naru.
I will become the god of the new world.
Although the translation "to become" is the same regardless of whether the particle used is と or に, there are nuances involved. I'm going to summarize here what I found on Maggie's site.
Using と with なる creates a more dramatic tone. It implies that the new state (which precedes となる) is a "final" stage, or something that takes a lot of effort to reach. In the case of Light Yagami in Death Note, becoming a "god of the new world" is an ultimate goal, and it will take a lot of effort to reach that goal.
On the other hand, using に with なる creates a more neutral, and hence less dramatic tone. It also has the connotation that the change in one's state is natural, or not extraordinary.
The use of the particle と with なる is not common in daily conversation mainly because of this dramatic nuance. For most cases, you use に with なる, as you have probably been taught in Japanese class.
Otona ni narimashita.
(I) have become an adult.
Raishuu, daigakusei ni naru.
Next week, I will become a university student.
(If one uses と instead of に in this sentence, this would imply that the person had a difficult time trying to get to university.)