This is ''The Creature from Cleveland Depths'', a short story by Fritz Leiber written in 1962. An engineer devises a little clockwork/tape-based thing where a man can make notes and reminders, and the device would give the wearer a slight shock, a "tickle" when the time hit, reminding him to check his appointments. His friend, a marketer, stole the idea (which he was noted as doing often, something the protagonist cheerfully grumbled about) and released it, claiming his company had been working on it all the time. As time went on, the tickler was improved to also automatically remind a person to do routine chores, and to even inject drugs to help make them happier. By the end of the story, the engineer, who has refused the tickler, realizes that the ticklers have taken over.
As per the details you provide above, the tickler sits on the shoulder and has a single eye. By the end of the story, they're rather obvious humps on those who wear them. And indeed, the ending has him reasoning with them that they can strike out on their own, and they travel out into space.
The scarred black tabletop was a dully gleaming silvery object about the size and shape of a cupped hand with fingers merging. A tiny pellet on a short near-invisible wire led off from it. On the back was a punctured area suggesting the face of a microphone; there was also a window with a date and time in hours and minutes showing through and next to that four little buttons in a row. The concave underside of the silvery “hand” was smooth except for a central area where what looked like two little rollers came through.
“It goes on your shoulder under your shirt,” Fay explained, “and you tuck the pellet in your ear. We might work up bone conduction on a commercial model. Inside is an ultra-slow fine-wire recorder holding a spool that runs for a week. The clock lets you go to any place on the 7-day wire and record a message. The buttons give you variable speed in going there, so you don’t waste too much time making a setting. There’s a knack in fingering them efficiently, but it’s easily acquired.”
Fay picked up the tickler. “For instance, suppose there’s a TV show you want to catch tomorrow night at twenty-two hundred.” He touched the buttons. There was the faintest whirring. The clock face blurred briefly three times before showing the setting he’d mentioned. Then Fay spoke into the punctured area: “Turn on TV Channel Two, you big dummy!” He grinned over at Gusterson. “When you’ve got all your instructions to yourself loaded in, you synchronize with the present moment and let her roll. Fit it on your shoulder and forget it. Oh, yes, and it literally does tickle you every time it delivers an instruction. That’s what the little rollers are for. Believe me, you can’t ignore it. Come on, Gussy, take off your shirt and try it out. We’ll feed in some instructions for the next ten minutes so you get the feel of how it works.”
....
Gusterson sucked in such a big gasp that he hiccuped. The right shoulder of Fay’s jacket and shirt had been cut away. Thrusting up through the neatly hemmed hole was a silvery gray hump with a one-eyed turret atop it and two multi-jointed metal arms ending in little claws.
It looked like the top half of a pseudo-science robot—a squat evil child robot, Gusterson told himself, which had lost its legs in a railway accident—and it seemed to him that a red fleck was moving around imperceptibly in the huge single eye.
....
“He’s only taking orders from himself,” Fay countered disgustedly. “Tickler’s just a mech reminder, a notebook, in essence no more than the back of an old envelope. It’s no master.”
“Are you absolutely sure of that?” Gusterson asked quietly.
“Why, Gussy, you big oaf—” Fay began heatedly. Suddenly his features quirked and he twitched. “’Scuse me, folks,” he said rapidly, heading for the door, “but my tickler told me I gotta go.”
“Hey Fay, don’t you mean you told your tickler to tell you when it was time to go?” Gusterson called after him.
Fay looked back in the doorway. He wet his lips, his eyes moved from side to side. “I’m not quite sure,” he said in an odd strained voice and darted out.
....
Fay was sitting as he’d left him, apparently lost in listless brooding. On his shoulder Pooh-Bah was rapidly crossing and uncrossing its little metal arms, tearing the memo to smaller and smaller shreds. It let the scraps drift slowly toward the floor and oddly writhed its three-elbowed left arm … and then Gusterson knew from whom, or rather from what, Fay had copied his new shrug.
....
Davidson pushed out from the wall against which he’d been resting himself and his two-stone tickler and moved to block the hall. But Gusterson simply walked up to him. He shook his hand warmly and looked his tickler full in the eye and said in a ringing voice, “Ticklers should have bodies of their own!” He paused and then added casually, “Come on, let’s visit your boss.”
Davidson listened for instructions and then nodded. But he watched Gusterson warily as they walked down the hall.
In the elevator Gusterson repeated his message to the second guard, who turned out to be the pimply woman, now wearing shoes. This time he added, “Ticklers shouldn’t be tied to the frail bodies of humans, which need a lot of thoughtful supervision and drug-injecting and can’t even fly.”
Crossing the park, Gusterson stopped a hump-backed soldier and informed him, “Ticklers gotta cut the apron string and snap the silver cord and go out in the universe and find their own purposes.” Davidson and the pimply woman didn’t interfere. They merely waited and watched and then led Gusterson on.
On the escaladder he told someone, “It’s cruel to tie ticklers to slow-witted snaily humans when ticklers can think and live … ten thousand times as fast,” he finished, plucking the figure from the murk of his unconscious.
By the time they got to the bottom, the message had become, “Ticklers should have a planet of their own!”
One of the other odd details that stuck in my mind was that the marketing friend drops off a case of lifelike masks for the engineer, prior products of the company, and he and his wife have a brief bout of fun while she's wearing the face of a prominent actress.
