The ye in Ye Olde was a printer's stand-in for "the"

Old English had a letter of its own for the sound th: þ, called thorn. It came from a rune of the same name. Scribes also had a second letter for the same sound, ð, and some used both. Thorn outlasted the other one and stayed in common use through most of the Middle English period.

In the 1300s the pair of letters th began to catch on, and at the same time thorn lost its tall upper stroke and grew less distinctive. By then it survived mostly in a few very common words and in shorthand. A scribe could write the as a thorn with a small e above it, and that as a thorn with a t.

Then came printing. William Caxton learned the trade on the Continent and set up his press at Westminster in 1476. He and the other early printers in England used type imported from Belgium and the Netherlands, and it had no thorn. It did have y, which looked similar, so the printers used that. The became ye, and that became yt.

Nobody pronounced the y. Even written ye, the word was said the. The first printing of the King James Bible, in 1611, uses ye for the in several verses, among them John 15:1, and yt for that in at least one. Later printings changed them to the and that.

Once th had replaced thorn for good, the link between the two letters was easy to miss. By 1896 the phrase "ye olde" was in use, to give a shop or an inn an air of Merry England. Today it is used more often as a joke, and the ye is often read like the old word for you.

Thorn is still in use in one living language. Icelandic kept it, and þ is the 30th letter of its alphabet.