This is pretty much an exception: it is indeed mostly pronounced as [니]. However, do note that it is technically [네] (dictionaries list this as the phonetic spelling), and it's only the common usage for you that is exceptional.
Why? For "I", we have 내가, which is similarly formed from 나. However, as most Korean learners can tell, 애 and 에 are not very easy to differentiate. Some even argue that there isn't any difference at all. Now then, that also means 내가 and 네가 would sound very similar, and confusing between you and me is a very inconvenient issue, so 네 has come to develop a more different pronunciation, that is [니].
How? The difference between the two vowels, is that 에 is more "closed" (more similar to /i/ 이), which easily supports that emphasis on /i/-like characteristics of the vowel led to 네 adopting the sound of [니] for common usage.
Note: 너 becomes 네가 rather than 너가 because historically the subject marker was 이, and it seems the contraction of the pronoun with the subject marker got stuck in everybody's heads and people keep using 네 although 너이 is no longer grammatical.