Here are my notes on the question, a summary of most important linguistic research with relevant quotes; perhaps someone, who is more into linguistic theory, might find them useful.
Hogg 1992 (2005):
- “Orthographic usage was reasonably stable during the Old English period”;
“Anglo-Saxon manuscripts of Latin normally kept to the Latin spelling conventions and alphabet, and in the earliest vernacular manuscripts the spelling system remained close to that of Latin texts, usually of Irish origin.”; thus, he continues,
“at first, we do not find spellings with <æ>, <w>, <þ> or <ð>” (p. 71; emphasis mine - Alex B.), Hoggs calls the use of <þ> or <ð>” the Anglo-Saxon innovations (p. 77) because in the very earliest manuscripts, especially from the north, we find the digraph th or the simple graph d (Hogg 1992/2011 mentions Bede(M), EpGl and the later LVD (p. 33);
- <þ> and <ð> “were in principle interchangeable with one another, whatever the generality of their usage or the habits of individual scribes” (p. 76), for instance, he quotes the following spelling variants from Beowulf, syððan (line 6), syðþan (line 132), syþðan (line 283), and syþþan (line 604).
- <þ> and <ð> “must have been used by the last quarter of the seventh century”;
- Differences in usage: at first, <ð> was the most frequent choice (up to about the time of Alfred); later, <þ> was more and more used, mainly restricted to initial position (with some occasional instances in medial position and rare in final position).
ð eth (or edh): “the native English term appears to have been ðæt” (Robinson 1973: 451); Hogg 1992: 75 adds that since the nineteenth century it has been known under its current name (Robinson 1973: 451 mentions its current name was borrowed from Modern Icelandic but Hoggs claims that “this is uncertain”).
digraph th: “appears to have been borrowed from Irish” (Hogg, p. 77);
Lass 1992 writes that “<ð> began to yield to <þ> in the thirteenth century, though it remains sporadically through the fourteenth” (p. 36) - against an incorrect assertion in one of the other answers posted here. He adds that in Early Middle English <þ> was the most common graphical representation for /θ, ð/, although the digraph th ‘reappears’ in the twelfth century and gradually ousts <þ>.
Upward and Davidson 2011:
“Through the ME period this TH was increasingly adopted in place of the OE symbols Þ and Ð to represent the native Eng voiced and voiceless sounds /ð, θ/” (p. 160). They list it under “Spelling changes introduced by Anglo-Norman scribes” (p. 176, emphasis mine - Alex B.).
Upward and Davidson 2011 mention that this gradual change was completed by about the end of the 15th century” (p. 176).
In passim, they make an accurate observation, almost a truism I'd say, about TH "being more economical and not susceptible to misreading as T+H" (p. 199, emphasis mine - Alex B.).
They add that “the digraph TH is not normally found in Eng words of pure Lat (as opposed to Graeco-Lat) or Fr derivation” (with some exceptions: faith, author, authority, anthem, and thyme) – see p. 199-200 for more details.
As I wrote elsewhere, drawing on Blake 2012, after the Norman Conquest, English monasteries started receiving a lot of monks trained in France who started copying OE texts applying spelling practices found in the languages they were familiar with (Old French and Latin?). Those practices were further spread by the other monks whose native language was English but who got trained primarily in Old French and a bit of Latin - Cecily Clark (Clark 1992) calls them the Gallicizing scribes.
In Scragg 1974 we can read that William Caxton, the first English printer, usually used the digraph th and only occasionally used a modified form of <þ>, the latter survived in private documents and in some printed books until the seventeenth century, esp. in the contracted forms of the definite article ye and demonstrative yt (p. 2).
But by that time th had already practically ousted <þ>, which had happened before printing.
For example, here's a passage from Raoul Lefèvre (translated by William Caxton), The Recuyell of the Historyes of Troye (1473).
cf. the following quote from Jensen 2012:
"The gradual loss of <þ> in favour of <th> in the English spelling system has traditionally been explained by the introduction of print. However, Stenroos (2006: 10) argues that the replacement of <þ> was a gradual process that had begun long before the advent of printing. She bases her argument on the following three points. Firstly, the use of <th> appears in the southern part of England already as early as ca 1300 and grows steadily more common through the fourteenth and fifteenth centuries. Secondly, Scragg (1974: 67) suggests that printing and scribal traditions may have been relatively isolated from each other until well into the sixteenth century. Thirdly, it is not completely obvious that the carrying over of thorn into printing would have posed an insurmountable problem: some types used in England did include thorn, and the letter ‘y’ was always available."
In Old English, this word was always spelled with the simple graph d. The PDE spelling with the digraph th is a Middle English innovation.