ある ‘some’ is an obvious rentaikei of the verb 有り, ‘to exist’, and its
emergence as a pronominal is due to the kanbun influence of Chinese 有 ɦuwX
‘same’ (Zisk, M. (2018). Middle Chinese Loan Translations and Derivations in
Japanese. Japanese/Korean Linguistics, 24. p. 323).

But what is と in the extended form, とある ‘some, certain’?

Martin, S. E. (1987). The Japanese language through time (p. 397). New Haven:
Yale University Press is silent on that. Iwanami Kogo Jiten just claims the
word exists and gives example, no etymology.
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etymology classical-japanese
What is the etymology of とある?
I'm almost 100% sure you mean 或る and not 有る.

A quick search in the デジタル大辞泉 gives:

> ［連体］たまたま行きあった場所や家、または日時などをさしていう。ある。「と或る食堂にはいる」「と或る夏の日のことである」

for [と或る](https://dictionary.goo.ne.jp/jn/154731/meaning/m0u/), and

> ［連体］《動詞「あり」の連体形から》はっきり名を挙げずに物事をさす語。また、漠然と物事をさしていう語。「或る所」「或る日」「或る人」

for [或る](https://dictionary.goo.ne.jp/word/%E6%88%96%E3%82%8B/#jn-7648).

If we go by this, と或る has an air of a something you just happened to come
across, while 或る is used to reference something without giving any identifying
details.

  * とある食堂に入る - to enter an eatery you just happened to come across 
  * ある食堂に入る - to enter a certain eatery (but let's not concern ourselves with the details)

Much like yourself, I wasn't able to find anything on exactly where this と is
coming from. However, because of the 偶然性 of と或る, if I had to reckon a guess
I'd say it may come from ふと, as in ふと思った.
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First of all, the meaning of 「とある」 with its fine nuances included would be "
** _a tidbit of_** ", " ** _of sorts_** ", etc. I specifically wanted to state
that before anything because by blindly believing the common "bilingual
dictionary" definitions of 「とある」, which are " ** _some_** " and " **
_certain_** ", one could unconsciously seek an etymological explanation that
would seem valid for those dictionary definitions. I think one would be left
in the dark if one did that.

To rephrase the above paragraph myself, the meaning of 「とある」 is closer to
「ちょっとした」 than it is to 「ある」 in Modern Japanese. The answer to your question,
however, is all in Classical Japanese.

Here is what I learned in high school Classical Japanese a few years ago. FYI,
I took no CJ in college, so you might want to wait for an answer by one of the
experts here.

「とある」＝「と」 + 「ある」

「と」 is an adverb in Classical Japanese. [Weblio
古語辞典](https://kobun.weblio.jp/content/%E3%81%A8+) gives the definitions
「そう」、「そのように」 and 「あのように」.

Originally, the adverb 「と」 was generally used in conjunction with another
adverb 「かく」, which means 「こう」、「このように」, etc. Thus, what we are talking about is
the 「こそあど」 here. Words/phrases such as 「とかく」「とにかく」、「とにもかくにも」, etc. should be
popping up in your head now.

"This and that", "like this and like that", etc. I stated 「とある」 actually meant
" ** _a tidbit of_** ", " ** _of sorts_** " at the beginning, didn't I?

In the 連体詞{れんたいし} (pre-noun adjectival) 「とある」, 「かく」 is not used but is surely
implied IMHO.

" **Existing this way or that way if not in a major enough way so it can be
named** " That would be my personal definition of 「とある」.
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