Can an illegal alien obtain citizenship through his father, a U.S. citizen, even though the father is now deceased

Your husband should review his facts with an attorney. Your husband's options really depend on the specific facts of his situation.

Offhand, it looks very likely that your husband was actually born a US citizen as his father was a US citizen at the time of the husband's birth. The law at the time of the husband's birth would likely be the controlling law.

There is no statute of limitation on a citizen's citizenship status. If born a US citizen, the person will remain a US citizen for life unless is stripped of US citizenship (for something like treason) or the person intentionally renounces his US citizenship (and a representative of the US government accepts the renunciation).

If your husband was born a US citizen, he is a US citizen regardless of how old he is now. He will need to submit an application with supporting evidence to the immigration service to get a certificate confirming his US citizenship.

Your husband should gather as much background information as he can and go see an attorney. Such background information include birth certificates for him and his father, other proofs of his father's citizenship, proofs of a relationshp between him and his father. The attorney will help him gather whatever other evidence is needed.