When did the Oriental Orthodox Churches start being called 'Orthodox'? The Eastern Orthodox (Greek, Russian and so on) are distinct from the Oriental Orthodox (Syriac, Armenian, Coptic and so on).

Do we know when the term "Oriental Orthodox" started being used for these churches as a group? Likewise, do we know when the individual churches started referring to themselves as "Orthodox"? I'm curious if the term "Orthodox" was first used in other languages (possibly by missionaries) and then later adopted by the churches.

I believe that the Syriac Orthodox, for example, only started using the word Orthodox fairly recently, in the 19th century.
The Greek adjective orthodox (ὀρθόδοξος) is dated to the late third/early fourth centuries, and the derived Latin word (orthodoxus or ortodoxus) was also first used at about that time. There is an earlier verb ὀρθοδοξεῖν, meaning "to have correct beliefs", used by Aristotle in the Nicomachean Ethics. As far as I know, Aristotle's coining is original, not based on any other language.

In Christianity, the term originally became popular to describe those groups that adhered to the definition of the Council of Chalcedon (451 AD), even though the Oriental Orthodox churches that you mention are exactly the opposite, not accepting the Chalcedonian dogmatic definition. The Oxford English Dictionary (under orthodox, 3) suggests that this is due to a later assumption that "Orthodox" refers to anybody who was not on the Catholic side at the time of the Great Schism (1054). So it seems that "Orthodox" was first associated with the Eastern Orthodox churches, and only later became applied to the Oriental Orthodox. 

For the reasons set out below, I think that the group term "Oriental Orthodox" was invented or at least popularized in the 1960s; and that at least some of the churches in question had Orthodox in their name for some years before that, though probably not earlier than the eighteenth century.

It is a bit hard to distinguish between a church that uses Orthodox as part of its name, and one that merely describes itself as orthodox. Certainly, people have been calling themselves small-o orthodox for centuries, and in some cases that shifted from "we are orthodox" to "we are the Orthodox". The unambiguous use of Orthodox in the name is comparatively recent, in all cases.

"Orthodox" as a title of the Eastern Orthodox churches

From what I can see, the term "Orthodox" was not used as a distinctive term until many years after the Schism. The more normal way was to talk about the Eastern Church and the Western Church. The everyday usage was "the Greeks", though this is of course inaccurate since not all Greeks are Orthodox and not all Orthodox are Greek. Meanwhile, Christians in general could be called orthodox or catholic for some centuries, regardless of their east-west allegiance.

For example, in the proceedings of the Council of Basel (1431-5), we read "the Greeks" as the standard term for the Greek Orthodox delegates, as opposed to "the Romans"; but "the orthodox faith" and "the catholic church" are both used to mean Christianity in general.

The Orthodox churches were certainly using the term as part of their identity by 1672, since the record of the Synod of Jerusalem contains many references to Orthodoxy as a tradition and as a communion (eg, talking about "our Orthodox religion", "Orthodox bishops", etc.). They were undoubtedly trying to use the term to distinguish themselves from the Roman Catholics, and to a lesser extent the Protestants. But even then, "Orthodox" was not part of the name of the church: the synod talks of "the Eastern Church", which holds "the Orthodox faith", but they don't use the title "Orthodox" for the church itself.

So at least in this period, the Orthodox placed additional emphasis on themselves holding to Orthodoxy as an ideal, in opposition to other groups. I suspect that the position of the Feast of Orthodoxy in the liturgy helped to establish this as a critical part of their identity. However, it seems that it may not have been used as a church title until a bit later. It was definitely established by the time of Nicodemus of the Holy Mountain, who used it in The Rudder (1800). Tentatively, I would say that it became used as a distinctive title at some point in the late seventeenth or early eighteenth century, though as set out above, there is a much longer history of the use of the word in the eastern Christian tradition. (The first OED appearance of "the Orthodox Church" is 1772, with "the Orthodox Confession" in 1679. Of course, that says nothing about non-English sources).

The Oriental Orthodox

As mentioned above, the application of "Orthodox" to these churches is quite unusual! There's even some evidence that the Eastern Orthodox deprecated the use of "Eastern Church" because they didn't want to be confused with non-Chalcedonian people (heretics!) who happened to also live in the east. Probably they were quite annoyed when those same groups started calling themselves Orthodox too. 

I would assume from the chronology that the individual Oriental Orthodox churches only started using the title a while after the Eastern Orthodox did. That would place the earliest use probably no sooner than the eighteenth century, and perhaps quite a bit later. Again, it may be that other people called them that, partly out of confusion as the OED suggests, before they called themselves that. So a 19th century date, in the case of the Syriacs, certainly makes sense.

Confusingly enough, "Oriental Orthodox" was also used as an exotic synonym for the Eastern Orthodox. The Oxford History of Christian Worship (Geoffrey Wainwright, 2006) says that the umbrella term "Oriental Orthodox" for the Syriac, etc., churches, was established in the 1960s, as a replacement for calling them "Monophysite". It certainly seems that pre-1950 references to "Oriental Orthodox" that I've found are actually talking about the Eastern Orthodox. Moreover, I imagine that the term "Oriental Orthodox" would only be invented if several of the churches in question were already calling themselves Orthodox.