1) "Let he who believes in this prophet speak now what he knows."
2) "Let him who believes in this prophet speak now what he knows."
To analyze this situation, it helps to separate the who clause (who believes in this prophet) from the main clause (let him/he now speak what he knows) and then analyze each clause separately.
Since let him speak now is correct, let him [who believes] speak now is also correct; and since let he speak now is not correct, let he [who believes] speak now is also not correct. Your second example is therefore the best one, grammatically speaking:
Ngrams shows that Let him who is also significantly more common than Let he who:
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Ok, let's look at this in terms of grammar. After all, the question asked which sentence is grammatically correct.
First, let's get rid of some words that may be confusing the issue. "Who believes in this prophet" is describing, or defining, he/him. It is a restrictive clause. Its only purpose is to define he/him. As a restrictive clause, it is a descriptive clause. We can remove it to figure out the him/he problem.
So what is he/him in this sentence? It is the direct object of the verb "let." As with any object, he/him is answering the question "what?" (for non-people) or "whom?" (for people). This gives us "Let what?" and "Let whom?" The answer to the question is an object.
Ok, so we need an object pronoun because the word is serving as an object of "let." What are the object pronouns? They are: me, you, HIM, her, it, us, them.
Look! "Him" is an object pronoun, so if we need an object, which we do in this sentence, we use "him." As such, "Let him who believes in this prophet speak now what he knows" is correct.
But wait a second! Isn't he/him the subject of "speak"? No. If that were the case, we would use "speaks," which is the third person singular form of "speak.' The sentence would read "he...speaks," but it doesn't here.
Now, the simple, practical solution: If we remove all the descriptive expressions from this sentence and pare it down to its roots, we get "Let HIM speak." I dare anyone to say "Let he speak" is correct.
Neil: Regarding speaker's judgment and usage: Sure, people can, and do, speak any way they choose. However, the question was about grammatical correctness. I'm not sure what "clause" you're referring to, but your example also demonstrates the point I'm making. In your example expression "They hoped for him to win," again, we see that "him" is serving as an object--in this case as the object of the preposition "for." We can ask, "For whom?" The answer will be an object.
If you need an object, whether the object of a transitive verb or the object of a preposition, your choice is "him," not "he." "He" is a subject pronoun; "him" is an object pronoun. This confuses many people, so I'll simplify: If you need the subject of a verb, use "he." If you need an object, use "him."