I was reminded of this curiosity when I read the following passage in one of the old Irish English emigrants' letters I use for my PhD research:
I am greatly obliged to you for yr [your?] regard
you express for me & I think myself very much
indebted to ye [the?] Person that has given me
so good a Character, if we ever have ye [the?]
Pleasure of meeting hope we shant be Disappointed,
for I assure you I promise myself a good deal of
happiness with you if we should ever see each other.
(S. Wylly, 20.06.1763)
This woman used the letter y several times in the word 'the,' but not always. A few times she writes 'the' and she also used th in words like 'brother.' I have not seen this used elsewhere in my corpus, but have not looked for it either. So where does this 'ye' instead of 'the' come from?
Probably from 15th century printer William Caxton. But first a little context.
'The' was written þe in Old English. The grapheme þ was used to represent both the voiceless sound in 'the' and the voiced sound in 'think.' This letter (called thorn) came from the runic alphabet fuþark.
|Upper and lower case thorn|
I turned to my colleague Kristian Rusten (who knows a whole lot more about Old English) for an example of thorn in Old English: On þæm æfterran geare Hannibal sende sciphere on Rome ('In the following year, Hannibal sent a fleet to Rome.' From the 9th century Old English translation of Orosius' 'Historiarum Adversum Paganos Libri Septem').
Today, only Icelandic uses thorn; but during the early days of the printing press (15th century), scribes (i.e. those copying texts by hand) in Britain still often used þ in English even though they used the Latin letters th for Greek theta, which is the same sound as thorn.
Getting back to William Caxton: Caxton was an English printer who started his business in Bruges. In 1476 he moved back to England where he introduced the art of printing. Chaucer's Canterbury Tales was his greatest success, though he was far from a perfect typographer. He often asked readers 'to correcte and amende where they shal fynde faulte.' Note that 'they' is spelled with th here, by the way... (Well, to be fair, yey would look ridiculous.)
Crucially for this story, his type punches (metal-cast letters used in the printing press) were made in mainland Europe and therefore did not include thorn because Dutch, German or Latin did not have that sound. In Britain, however, Caxton did need to print the sound used in 'the.' As he did not have a thorn letter punch, he sometimes used th (as used for Greek theta), but sometimes also y because it looked the most like the grapheme þ. (Spelling conventions were still a bit fluid at the time.) Of course, in Helvetica (the font you are reading now) y does not look like þ very much, but in the blackletter type used by Caxton (think Gothic) it did. And so 'ye' for 'the' was born.
In short then, y in the spelling ye ('the') is simply the result of needing to find the closest-looking equivalent of þ in an era when th wasn't yet the unchallenged spelling, and has survived in fixed phrases such as 'ye olde shoppe' because... it looks... old?
So, 'ye olde shoppe' should really just be pronounced 'the old shop.'
Oh, if you do open that shop(pe), tell the sign maker to use thorn and I will definitely come by and buy something.
 Baron, Naomi S. 1988. 'From Printshop to Desktop: Evolution of the Written Word.' Georgetown University Roundtable on Language and Linguistics. Baltimore: n.p., pp 8-21.
 Garfield, Simon. 2011. Just My Type: A Book About Fonts. London: Profile Books.
 Clemens W.A. Fritz. 2007. From English in Australia to Australian English, 1788-1900. Frankfurt am Main: Peter Lang GmbH.