Given that the two "th" sounds don't actually sound like a combination of "t" and "h" why was that particular combination selected or become adopted by the majority ?
The "TH" being pronounced as an interdental fricative is called digraph, which is a "pair of characters used to write a single phoneme" (a single sound, said in plain words).
EDIT: @z7sg and @mP01; I found the historical course of this sound/letter:
Far too many manuscripts from ancient Roman times are lost to know for sure who chose TH to represent θ and why. It wasn't the /θ/ sound we think of now, though.
The reason it was used in the first place was that after the Normal conquest, French became the primary language of the literate, and "th" was the closest that French had to the sound. Nowadays, "t" and "th" are the same in French, but a thousand years ago it was still a somewhat different sound and could maybe substitute if you didn't mind sounding foreign. So that was the natural choice to transcribe our words their way, even if it wasn't /θ/, without resorting to weird pagan runes.
In other words, it was because the first scribe decided to do it that way, everyone else followed the same pattern, and it wasn't important enough to ever bother changing.
It won out over the thorn "þ" (later simplified to "y") in Elizabethan times largely due to French being the refined language of the nobility and upper classes, so it was kind of inevitable that it would percolate down to the lower classes, especially when the upper classes drove most of the printing that was unifying English around the country. The "French" commonly spoken in England was more of a branch from medieval French mixed with some Parisian, becoming its own distantly related language before merging entirely into English, but it was still seen as superior to English long after it was essentially dead. (This is now called Anglo-Norman French, and later Law French. Many current legal terms come from this offshoot of French.)
This is the same reason we have silly grammar myths like "no split infinitives" -- they just can't exist in French, so they obviously shouldn't in English. The whys were slowly forgotten, but the rules persisted.
You can still see thorn as "y" in early American colony documents, but by the revolution it had completely disappeared. The Francophilia of many upper class Americans certainly helped there, too.
Let me offer my personal speculation, for what it's worth of course.
But first, I need to introduce a fact well known to students of Old English but that probably needs to be mentioned here.
The vast majority of preserved Old English written documents come from Wessex. King Alfred was the most prominent king of the House of Wessex but the whole period between the 6th Century and the Norman invasion is dominated culturally by the Kingdom of Wessex.
The dialect spoken in Wessex and Sussex was West Saxon whilst Mercian was spoken in Mercia, East Anglia and Essex (including in London), Northumbrian in Northumbria and Kentish in Kent.
After the Norman invasion, French was ubiquitous but Middle English gradually recovered from London. That is, with a strong Mercian East Midland background.
Now to the point.
The initial English spelling of /θ/ was indeed th as in Latin. My understanding is that the 'þ' (thorn) and 'ð' were introduced in Wessex where they replaced the initial 'th' inherited from Latin, whereas other kingdoms of the Heptarchy were probably immune to this trend (that's what needs more investigation).
Later when the evolution of English was driven from London, the use of 'th' came back into fashion, also probably under the growing influence of Latin (non Latin characters being probably considered less prestigious).
wikipedia has the following to say about 'th':
The Saxon origin is right, the Norse isn't. Here is how things came to be.
Here is a last citation from wikipedia regarding this part of the story:
The modern digraph th began to grow in popularity during the 14th century; at the same time, the shape of thorn grew less distinctive, with the letter losing its ascender (becoming similar in appearance to the old wynn (Ƿ, ƿ), which had fallen out of use by 1300) and, in some hands, such as that of the scribe of the unique mid-15th century manuscript of The Boke of Margery Kempe, ultimately becoming indistinguishable from the letter Y. By this stage th was predominant, however, and the usage of thorn was largely restricted to certain common words and abbreviations. In William Caxton's pioneering printed English, it is rare except in an abbreviated the, written with a thorn and a superscript E
Now why one might be under the impression that the thorn came from Norse is that it was used in Old Norse around the same time as Old English and survives till today in Icelandic. It is supposed that the thorn in the Icelandic alphabet is also from Runic origin.
influence alone can not be held responsible for the gradual fall out of use of the
It's a question of how printing began in Britain. In written script even late in the middle ages thorn þ (usually) or eth ð was used; "th" was found bot wasn't the most common.
But printing started on the continent and the first English printers bought their typefaces from there. Since the 'th' pronunciation was rare on the continent (and not seen in Latin which a great many books were written in), there were no typefaces for thorn þ or eth ð, and English printers were forced to use "th". Since printing produced vastly greater numbers of books and documents than scribes could, the "th" spelling became the most common and replaced the old letters.
"Ye" as in "Ye olde shoppe" came about due to misunderstanding old manuscripts hands which, written quickly often didn't close up the top of thorn so that þ often looked like Y.
TL;DR: it's more or less a convention (going way back) that an "h" changes the preceding consonant from a plosive into a corresponding fricative.
I.e. you simply loosen up your lips and get "f" (ph) instead of "p," and "th" instead of "t."
Having "th" stand for both "th" and "dh," in my humble opinion, is a historical mishap.
The origin of 'th' really has nothing to do with the development of the English language. It comes directly from the Roman alphabet.
It is likely that this combination was chosen by an early scholar while translating Greek and this standard was then generally adopted. The 'th' digraph can be found, for example, in the writing of Lucius Livius Andronicus (c. 280/260 BC – c. 200 BC).
According to the OED: