The name originates only in the second half of the 20th century. OED and Etymonline both say the earliest reference to the term comes from 1960. Indeed, OED's list of notable citations shows how they went from exotic in the 1960's to fairly normal by the 1980's in North America:

1960 M. G. Smith et al. Ras Tafari Movement in Kingston, Jamaica ii. 13 The plaiting of long hair by men known as the ‘men of dreadlocks’... These men of dreadlocks were the Ethiopian Warriors and the self-declared Niyamen.

1966 Guardian 22 Apr. 13/7 An extremist fringe of members distinguished by their long twisted hair or ‘dreadlocks’, which were modelled after pictures of Ethiopian warriors.

1971 A. King One Love 39 He..is a Dread-Locks Rastaman, who, to most people, appears as a strong half-Chinese who has neglected to cut his hair.

1975 Globe & Mail (Toronto) 11 June 3/5 Many [Rastafarians] wear their hair in long, matted wool-like ringlets or plaits called dreadlocks.

1977 Sunday Times 26 June 52/6, I have always since felt a sympathy with these dreadlocked believers in the divinity of Haile Selassie.

1980 Daily Tel. 13 Nov. 2/2 Employers could be influenced by young blacks who came to an interview with the long ‘rasta’ dreadlock hairstyle.

1981 Times 8 Sept. 2/7 Lord Scarman..told the inquiry that the image of the young Rastafarian with dreadlocks was based on the Masai warrior.

1984 New Yorker 26 Mar. 47/2 He had a Dodgers cap on top of his dreadlocks.

Dreadlocks' popularity. as far as I understand them, come at the confluence of two movements, one of which I understand much more than the other. Dreadlocks have theological significance in Rastafarianism. Rastafarianism is a difficult religion to study, and one that I wish I knew more about. It was traditionally loosely organized in Jamaica into three "mansions" (think, perhaps, sects or movements): the Bobo, the Twelve Tribes of Israel, and the Nyahbinghi, but there are have been countless splits and individual practitioners unaffiliated with any of the main movements. Though the Christian Bible is deeply important to Rastafarians, there is a tremendous amount of oral tradition that makes it more difficult to study than many other New Religious Movements.

It's unclear when exactly Rastafarians started wearing dreadlocks. The movement started in the 1930's, and the earliest point I've seen from a historian is the late 1940's. Barry Chevannes, the Jamaican social anthropologist and oral historian perhaps most associated with the academic study of Rastafarianism, suggests that it emerged perhaps as early as 1949, but building off earlier attempts to set themselves apart sartorially (the movement only starts in the 1930’s). While the general story is that Jamaican Rastafari were inspired to grow dreads from "Ethiopian warriors" (perhaps actually Masai or Mau Mau), Chavannes argues that it grew out a particularly fanatical early sect of Rastafarianism, the Youth Black Faith (who apparently later formed the basis for the Nyahbinghi Mansino), and spread from them spread to the movement more broadly. His chapter "The Origin of the Dreadlock" is unfortunately not available online, so I am relying on a summary of Chevannes's work in a chapter from Michael Barnett's Rastafari in a New Millennium: a Rastafari Reader (pg. 228):

Chevannes (1998b, 1998c) suggests that the first group of Rastafarians to wear dreadlocks was a group of young dissident Rastas known as the Youth Black Faith (YBF), which emerged in 1949. According to Chevannes, this group consisted of younger Rastafarian converts who were interested in changing the revivalist practices within the organization. It is Chevannes's contention that the YBF impetus for wearing dreads did not emerge from East Indian prototypes or, as suggested by Campbell (2007), from images of Masai warriors, but from a desire to be viewed as deviants.

Before Rastas emerged on the scene there were homeless men who roamed the streets with long, unkempt hair that was essentially the same as dreadlocks worn today. They were unkempt in hair, dress, and cleanliness and were regarded as outcasts by the rest of Jamaican society. This practice is similar to that adopted in the early 1930's, when Rastafarian men wore beards to more closely associate themselves with Haile Selassie and to distinguish themselves from the general populace. Even though there have been women Rastafarians since the beginning of the organization, the group was often referred to as "bearded men, the beards or beard men" (Cheyennes 1998c, 98).

This sartorially double marking of separateness and piety is common among religious groups: the Sikh turban, the Jewish yarmulke, the Muslim hijab (neither here nor there but Iddo Tavory has a great sociological article on this called “Of Yarmulkes and Categories: delegating boundaries and the phenomenology of interactional expectation”).

The "dread" is from fear, but it's unclear if the dread comes from that these men were inspiring or that they were showing that they "feared God".

One theological complication is that, over time, many Rastafarian practices have been modeled on Old Testament "Nazarites" who took a vow which, among other things, precluded them from cutting their hair or drinking wine (Samson is probably the most famous Nazarite in the Hebrew Bible). To this day, many pious Rastafarians will abstain from both cutting their hair and drinking all alcohol (some others abstain only from wine). Some groups believe that all Rastas must take the Nazarite vow, whose sign is normally taken to be dreadlocks (not cutting hair), and not all groups hold that that is necessary (there's a pretty recent Morgan Heritage song called "Don't Haffi Dread", whose refrain begins "You don't haffi [have to] dread to be Rasta").

The Rastas, particularly through reggae music, made this style of hair very popular throughout the African diaspora. As little as I know about Rastafarianism, I know less about the other major movement that helped spur dreadlocks' popularity: the natural hair movement, which was a part of the broader black power/afro-centricism/black is beautiful movement. Perhaps all I can suggest is Chris Rock’s excellent documentary, Good Hair. For most of their time in America, African-American beauty standards were based on European-American beauty standards. In the 1970's especially, a "Black is Beautiful" movement started to encourage people--especially black women--to take pride in their "natural" looks. It's no coincidence that the afro, braiding, and only slightly later the dreadlocks started becoming popular at roughly the same time, from the 1970's onward. The Wikipedia article called "Afro-textured hair" gives some more detail, though I would be wary of trusting every detail there. The pictures are amazing, however, and show the increased range of hair styles that “Black is Beautiful” Afro-centricism afforded African Americans, especially African-American women.

Many in the Black Diaspora adopted the Rastafarian dreadlocks--including the name--without adopting its theological valence, which had come to be closely associated with the Nazarite vow (I believe in all Rastafarian traditions, but I’m having difficulty even confirming that). M. Bahai Kuumba and Femi Ajanaku have an interesting articled "Dreadlocks: the Hair Aesthetics of Cultural Resistance and Collective Identity Formation". They use a very neo-Gramscian via James C. Scott framework of "hegemony and everyday resistance", which was unfortunately very fashionable at the time this article was written (Marshall Sahlins has a few great pieces in his Waiting for Foucault, Still about the intellectual poverty of framing everything as either hegemony or resistance). For them, dreadlocks represent a social movement, though one that had lost its potency in the mid to late 1990s as the style had become more popular. They categorize the spread of dreadlocks into four "somewhat overlapping sociohistorical phases": 1. the Rasfarian phase (mid-1950's to the mid-1970's), 2. the African nationalist/Pan-Africanist fringe phase (mid-1970's to mid-1980's), 3. the "broader counter hegemonic phase" (mid-1980's to early 1990's), 4. the transitional phase (early 1990's to the publication of the article in 1998). As of the writing, it was not quite clear what the transitional phase was transitioning to, but I took it to mean an apolitical style.

Until the mid-1980's, dreadlocks were only popular in small "pockets" of the Black Liberation movement. Increased political activities, in the late 1980’s and early 1990's, they argued spread the hairstyle it became "counterhegemonic practice in a broader sense". By the mid-1990's, they say, dreadlocks are both "commercialized and criminalized" in the media through their association with apolitical music and high fashion on one hand and drug dealers and gangs on the other. Where the style would go then bumps up against both the publication of this article, and our “20 year rule” here.

That's mostly the end of the history section, but a further discussion of "cultural appropriation" continues below.

My question sounds slightly loaded, I should be more clear about my definitions. What I mean by "distinct history" is a history that has ties to a specific people, religion, or time period.