My friend is in USA on H4. He had applied for H1b and got an approval. Can he apply for SSN now. If not, when can he apply? What are the documents required?
Thanks.

Yes, apply to the Social Security Administration. Go to their website for details.

He has applied for ITIN but have not yet received it. Can he apply for SSN while his ITIN application is still pending?

Yes, he can apply for the SSN. The ITIN is a temporary solution which will be superceded by the SSN.

He had gone to the SSA office with all the documents and I797. But, he was told that its too early right now to apply since his H1B status starts in October and he needs to come back later. His application for SSN was not accepted. Can you tell me which month should he now go to SSA office to apply for SSN.

I would apply is late July/early August.