Why was the "th" combination chosen for the "th" sound?

The origin of 'th' really has nothing to do with the development of the English language. It comes directly from the Roman alphabet.

From Wikipedia.:

The digraph ‹th› was introduced in Latin, which used it to transliterate the letter theta ‹Θ, θ› in loans from Greek. Theta was pronounced as an aspirated stop /tʰ/ in Classical and Koine Greek.

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:

The Greek letter, which corresponds etymologically to Sanskrit ध dh (and so, by Grimm's Law, to Teutonic and English D), was in early inscriptions represented by TH, and was a true aspirate;

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':

th: Represented /θ/ in the earliest texts but was soon replaced by ‹ð› and ‹þ›. For example, the word meaning "thought" was written ‹mōdgithanc› in a 6th century Northumbrian text, but later as ‹mōdgeþanc› in a 10th century West Saxon text.

and

þ: An alternate symbol called thorn used instead of ‹ð›. Represents /θ/ and its allophone [ð]. Together with ‹ð› it replaced the earlier ‹d› and ‹th›. First attested (in definitely dated materials) in the 8th century. Less common than ‹ð› before Alfred's time, from then onward ‹þ› was used increasingly more frequently than ‹ð› at the beginning of words while its occurrence at the end and in the middle of words was rare. Some modern editions attempt to regularise the variation between ‹þ› and ‹ð› by using only ‹þ›.

Important Edit
This update is intended to answer two similar questions (see comments below) which have raised the possibility of the letter thorn (and others such as ðæt ‹ð› wynn ‹ƿ›) actually being originated "from the Norse or Saxons".

The Saxon origin is right, the Norse isn't. Here is how things came to be.

1. In Roman Britain, texts were written using the Latin alphabet. Which is fair enough since they were Latin texts. This was also the time when the Christian religion first entered Britain because it had become the Late Roman empire's official religion.
   Greek words including a theta (θ) were already transliterated in Latin using a 'th' digraph.

2. When the Saxons invaded England, they came not only with their language but also with the alphabet that was used to put it in writing: the runes. The runic alphabet included such letters as the thorn ᚦ and the wynn ᚹ. It was an alphabet remotely inspired by the Greek alphabet probably passed on by the Etruscan with whom the Saxons were trading. Only very few inscriptions of Old English in Runic alphabet have come down to us.

3. These Saxons were pagans and the Catholic church undertook to Christianise them. As a result of this effort, Latin made its come back in England together with the Latin alphabet. The texts were mostly Latin but some included Saxon words transliterated in Latin . However there was no character for the English th sound in the Latin alphabet and two different characters were borrowed:

   • the ðæt (ð) because this is how the Irish monks had already solved the problem (Ireland had not been invaded by the Saxons and it had never ceased to be catholic, had always used the Latin alphabet and some of its monks were enrolled along with the ones who had sailed over from Rome to Christianise the heathens).
     Since the th was similar to a d, it was noted as a d with a cross-stroke added cutting its top ascender (a common habit also used for other letters).
   • the thorn (þ), because that was how it was noted by the Saxons themselves using the Runic alphabet.
   
   So that these two letters gradually came to be used interchangeably in Old English to note the th sound. Even in the same sentence you could find the same word using either one or the other.

4. In a few decades, the Vikings had managed to kill a significant proportion of the Latin speakers in Britain and only Wessex had been spared thanks to King Ælfred's mix of stength, luck and diplomatic abilities.
   Having secured his kingdom, and now waiting for the Danelaw to fall ripe, this very able king nearly single-handedly managed to rekindle the once vigorous English cultural life. With one change though: for want of Latin scholars, the majority of texts were now written in Ænglisc (which we now term Old English).
   Quite naturally the runic thorn and wynn along with the Irish eth and a few other exotic 'letters' were used to write this nascent English literature.

5. That is, until an arrow landed in Harold Godwinson's eye.
   There are however more than a single reason why all the non standard Latin letters fell out of use.

   • Sure the Anglo-Norman scribes had not learned these letters and were reluctant to use them. But they were only writing French anyway.
   • Mercian London rather than Wessex Winchester was now the beacon of cultural and economical life.
   • Latin was not just the ecclesiastical language any more but also the scholarship language.
   In fact it took more than 200 years for the th digraph to regain popularity after the Norman conquest.

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.

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 Norman 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 creole 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.