Where does "on X's dime" originate from?

Phrases like "on the company's dime" or "on your parent's dime" are used to indicate a form of payment responsibility on the party in question.

My question is where does "on X's dime" originate from? Why was the word "dime" specifically used as opposed to say "on the company's quarter/nickel/penny"?

The earliest Google Books match for "on [someone's] dime" that I've been able to find is from Anna Richardson, "Who Gets Your Dime?" in McClure's Magazine (November 1916), where the dime refers to the price of a movie ticket:

The next investment made by the producing-firm, and one which draws heavily on your dime, is salaries. These too are variable, but in the aggregate they represent amazing figures.

A second instance is from Irwin Shaw, Sailor Off the Breman, and Other Stories (1939) [combined snippets]:

"I got a date," he said aloud, savoring the words. He went to the shade of the grape arbor to think about it. He sat down on the bench under the cool flat leaves, and thought about it. He'd never had a date before in his life. He had thirty-five cents. Thirty-five cents ought to be enough for any girl, but if he hadn't bought the radish seeds, he'd have had forty-five cents, really prepared for any eventuality. "Damn crow," he said, thinking of the evil black head feeding on his dime.

So in the pre–World War II era, we have the dime in "on [someone's] dime" referring to the price of a movie ticket and to the price of a packet of radish seeds.

In the decade after World War II, several instances of "on [someone's] nickel" appear. From J.R. Tranter, letter to "Dear Editor," in Sales Management, volume 63 (1949) [combined snippets]:

I have lived in Lansing since 1934—and since that time have bought $300,000 worth of life insurance; two automobiles each year when autos were procurable; refrigerators, radio, two residences. ... The other day I made a phone call about service on a new disposal unit I had bought. On my own nickel, the person answering the phone asked me if I had bought a television set yet.

From Perry Githens, "How to Listen to Science," in Popular Science (September 1949):

Don't worry about the words in this or any other issue of POPULAR SCIENCE. Some of them are new to scientists, too. Just lend an ear, because Public Science is playing on your nickel, your life—and ours.

And you own the juke box.

And from Polly Adler, A House Is Not a Home (1953):

The first five minutes of our telephone conversation was taken up with a blow-by-blow description of his personal success story and what a big shot he had become. I wasn't impressed and I implied that if he wanted to beat the drum for himself, okay—but not on my nickel. So he got down to business.

The first and third instances refer to the price of a phone call; the second to the price of playing a song on a jukebox.

Conclusion

As Josh61's answer points out, Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) ascribes the expression "on [someone's] dime" to the cost of a telephone call:

on someone's dime (or nickel) adv phr At someone's expense other than the speaker's: On the dime of Universal—G[entleman's] Q[uareterly, date not given]/ Hicks and Sims went home, on their own nickel, to their own miseries—Esquire [1994] {fr the dime or nickel formerly needed to activate a pay telephone}

The Google Books search results I've cited call this attribution into question, however, and suggest that the phrase may have arisen more broadly in connection with anything that cost a nickel or a dime. Moreover, the expression is not limited to "someone's expense other than the speaker's," as multiple instances of "on my dime [or nickel]" and "on my own dime [or nickel]" indicate.

This is the only reference I could find that supports the clear explanation given by Robusto:

On someone's dime:

• At someone's expense other than the speaker's : On the dime of Universal/ Hicks and Sims went home, on their own nickel, to their own miseries

  • The expression is from the dime or nickel formerly needed to activate a pay telephone.

The Dictionary of American Slang, Fourth Edition

The expression appears to have been used from the mid 60's as shown in Ngram: on my dime, on his dime.