What are the required properties of a PRNG to make it suitable for a stream cipher? As an example, if a PRNG passes the BigCrush test suite, and has a period of 2^1024, can it be used on its own to make a secure stream cipher?

What are the required properties, and similarly, what are the bad properties that the PRNG must not have?
The best known algorithm to guess the next bit with probability near 1, without knowing the $n$-bit seed, has to cost approximately $2^\lambda$ bit operations, where $\lambda \leq n$ is the security parameter.

You can't determine this by feeding the output of the PRNG to a generic statistical hypothesis test, because by its very nature that statistical hypothesis test has only very limited hypotheses about how the PRNG might differ from a uniform random string of bits.  These generic statistical hypothesis tests are designed without particular knowledge of the PRNG.

You can really determine this only by feeding it to a swarm of cryptographers, but that works only if it appears as tasty to cryptographers as a bleeding hand to a piranha.

Cryptographers are a picky bunch that aren't much interested in spending their time on designs that aren't really interesting for some reason, like protecting trillions of euros of financial transactions, or performing better than modern standard stream ciphers today (like ChaCha, or AES-CTR with hardware acceleration) at the same conjectured security level.

And they don't usually want to spend time on designs whose designers haven't already studied and demonstrated to resist a litany of standard attack strategies.