I have thought Anglican prayer beads to be an adaption from the Eastern Orthodox chotki. I gravitated to them readily since I had an earlier flirtation with Orthodoxy before becoming Anglican. I wore out my knotted chotki a lot faster than Anglican beads. I suspect they may also be very popular among Catholic converts to Anglicanism who were
experienced with praying the Rosary.
Are their roots Orthodox or Catholic? Well, it Depends.
I somewhat prefer the Lestovka to the Chotki, particularly the cheaper chotki, which tend to feel rough when handled (although this can be mitigated. Due to the Church of the Nativity in Erie, PA inadvertently duplicating an order I placed for two vinyl Lestovkas after I misplaced one, I now have several of the Old Rite variety (I offered to send the extra items the Church sent me right back, but they graciously told me not to worry about it, so I donated the two extra copies I had of the Horologion to my local ROCOR parish for their bookstore, but because lestovkas tend to wear out, and also someone might need one), and one very nice leather Lestovka configured for the Prayer Rule of St. Seraphim of Sarov (which I purchased from some OCA nuns for just $10, usually they are considerably more expensive), which involves 200 Ave Marias in the Orthodox version, which has slighlty different wording, divided into groups of 50, with no Decades, no mysteries and no visualization of those mysteries, but it can also be used for the Jesus Prayer. In general, leather lestovkas are worth the added expense, and so I was greatly blessed. The Old Rite form has an impressive number of different counters, some for the Jesus Prayer, some for Kyrie Eleisons during the Hours, some for the Prayer of St. Ephraim said in Lent repeatedly, a specific number of times in the Russian Old Rite, and some for prostrations, but because of these, you can also use it to perform multiplication and implement arbitrary prayers.
I do not have any Anglican prayer beads in my collection, but I would really like to add some, and have only one Roman Catholic rosary.
By the way, I take these prayer ropes and related devices seriously and endeavor to use them as much as possible, although unlike some people I don’t find that I need them; it is an accident that I have as many as I do, which is to say, too many, but not an accident that I have a diverse set.
I think a major use for them in someone who otherwise would not require them is to use them counter-intuitively; they can induce a temptation to pray more rapidly, which should be resisted; if used extremely deliberately they can assist one in praying at a pace which appropriately reflects the duration of the prayers. I dislike hearing people rush through reciting forty Kyries, which is a thing in those Orthodox churches using the traditional Sabaite typikon, where Terce, Sext, and in Lent, Noone, are served before the liturgy, and also in the Coptic church (where forty one kyries are recited). Obviously this is not a problem one would normally encounter with the Anglican Rite, which despite silly accusations from the Puritans, is, among traditional liturgies, extremely resistant to that particular problem.
By the way, this takes me to a segue which some of you might find interesting: if anyone ever criticizes Anglicanism for a lack of liturgical antiquity, this is untrue, and can be disputed on this basis: the current recensions of the liturgy in most churches are approximately the same age as the Cranmerian liturgy. For various reasons including the Turkish conquest of Constantinople, the Council of Trent, and the Nikonian reforms, and coincidentally, concurrent events in most of the Oriental Orthodox, almost all traditional liturgies in their current configuration are approximately five hundred years old. There are a few exceptions, for example, the Carthusian mass in its pre-Vatican II state has not changed substantially since implementation in the 12th century, and the Russian Old Rite is about the same age (although Old Rite parishes are celebrating it in a very meticulous way which reflects rejection of the Nikonian reform of the liturgy in 1660). Most Greek Orthodox churches are using a typikon from the 19th century, revised in 1890, and then tweaked again to accomodate the Revised Julian Calendar, with the exception of the Old Calendarists, the monks on Mount Athos, and the Church of Jerusalem (and the autonomous Church of Sinai, which consists of the Monastery of St. Catharine and little else). So the age of the 1662 BCP fits right in, interestingly enough.
Also, the historic BCP / Anglican Rite is a recension (and a brilliant and elegant simplification) of the old Sarum Rite, whose implementation of tje Divine Office was inspired by a proposed revision of the Roman Breviary by Cardinal Quinones, with certain explanatory texts by Cranmer for the sacramental services (which often begin with “Dearly Beloved Brethren”) and also some borrowing from the Byzantine Rite (specifically, the Prayer of St. Chrysostom, which is the Prayer of the Second Antiphon except in the disused 1890 recension of the Divine Liturgy of St. Mark, where the Liturgy of the Word differs from that of St. Basil and St. Chrysostom only in respect to the prayers of the antiphons, and also the Epiklesis adopted by the Non Jurors from the Divine Liturgy of St. James, (which one can occasionally find in use on October 23rd, his feast), and passed on to the Episcopal Church USA and thence to most of the other Anglican churches in the US, for example, the Continuing Anglican churches).