Hello, I am not a historian, but I am a linguist. I do not know if any ancient civilizations made any conscious effort to develop sign languages, but there is good reason to think that sign languages existed, and maybe some modern evidence for spontaneous generation of sign languages could shed some light on this question? All of the following assumes that the human brain has not substantially changed over the last 50,000 or so years, which seems like a safe bet.
There do exist limited examples of pre-western sign languages, such as Plains Indian Sign Language spoken already when Europeans first arrived in the New World, but the records going further back do not exist to my knowledge --- with the exception of Plato's Cratylus, where Socrates says ''If we hadn't a voice or a tongue, and wanted to express things to one another, wouldn't we try to make signs by moving our hands, head, and the rest of our body, just as dumb people do at present?"
While the principle sign languages most Westerners are familiar with, American Sign Language and so on, were based on a signed language made expressly with the intention of providing deaf people with a way of communication, there is evidence that communities of deaf people will spontaneously create sign language which are different only in modal terms (that is to say, use the same cognitive resources) from spoken languages.
First, it has been noted that babies undergo a similar linguistic developmental process regardless of whether or not they are learning spoken, or signed languages. This is suggestive that babies are hardwired to use gestures as language if they do not acquire spoken language. Since most linguists agree that language is innate (though, as many probably know this is not universally agreed upon but is certainly mainstream), if we accept this premise, then babies are perfectly capable of innately generating signed languages.
Even if this premise is not accepted, we have actual evidence for the development of full fledged signed languages. The most well-known example comes from Nicaragua, where deaf children who previously had lived isolated lives were put together in pensions with the intention of giving them an education, ironically mostly using spoken Spanish. What ended up happening, however, was that the children developed their own system of signs. This generally is not considered a true language, because the children were constructing an ad hoc system of signs that didn't really work like a natural human language. However, and crucially, the younger children who inherited this system of ad hoc signs DID use it in a way that corresponds to what we know about natural human language. Thus a new language was born, a spontaneous sign language.
There are other examples from smaller communities, such as a case among a small Bedouin community in the Negev. This shows that the community of deaf speakers need not be exceptionally large for sign languages to be developed.
In fact, the urge to communicate, via speech or gesture, seems to be so strong that deaf children will do it even if they don't have other deaf children to talk to.
While I imagine that historians do not like to draw firm concrete conclusions when there is no historical record to base such conclusions on, the evidence that the urge to create sign language is so strong that ancient people almost certainly did is quite strong.
Edit: I see probably similar posts have already been made making the same points. Delete as needed.
There is a line from Plato's Cratylus which provides evidence that there was/were sign language(s) in ancient Greece. In this dialogue, Socrates is debating with Hermogenes regarding the origin of names. Socrates believes naming is an art, and that good names should reflect the essence of their subject while the worst names do exactly the opposite. On the other hand, as Socrates asks of Hermogenes:
Hermogenes: let us see;- Your meaning is, that the name of each thing is only that which anybody agrees to call it?
That is, Hermogenes believes names are purely conventions.
Anyways now that the context of the dialogue has been established, here is the line:
Socrates: Very good; but then how do the primary names which precede analysis show the natures of things, as far as they can be shown; which they must do, if they are to be real names? And here I will ask you a question: Suppose that we had no voice or tongue, and wanted to communicate with one another, should we not, like the deaf and dumb, make signs with the hands and head and the rest of the body?
Emphasis added; that last part seems to indicate the deaf and dumb of Greece at the time did use signs. Admittedly, this isn't strict evidence that such signs constituted a sign language, in the strict linguistic sense of the term. You can have systems of speech or sign that do not constitute a language (pidgins are the best example). But, for reasons /u/ostuberoes has already enumerated, the organic development of sign language is pretty well attested in modern times; in fact, very young children seem to be capable of taking a system of signs that is not a language and developing a language out of it, in the same way children raised in pidgin speaking environments develop a complete language (a creole) out of it. Again, see the aforementioned user's comment for the evidence of this. So it seems reasonable to believe that the signs Socrates refers to are, in fact, a sign language.
Sources
A note to commenters: If you're here for an answer, there isn't a sufficiently good one posted yet. The thread is highly upvoted because people are interested in the question, not because there's an answer. If you think you have an answer, that's great! Before you start writing, though, take a spin through our rules on answers and make sure you can write something in-depth and comprehensive that addresses the (general lack of) historical record of sign languages, and how we can know an answer to this question despite that.
If your only contribution is to ask about deleted posts, or why there's no answer, you're very likely to be temporarily banned. If you have a complaint about moderation, start a meta thread or send us a modmail.