First of all, this is a show only event. It has never happened in the books as Books ended when the Dothraki found Daenerys. Later, in absence of the Queen, Ser Barristan Selmy assumed the office of Hand of the King to govern the realm. From ADWD, Chapter 59:
Though he had assumed the title of Hand, Ser Barristan would not
presume to hold court in the queen's absence, nor would he permit
Skahaz mo Kandaq to do such.
However since you have added the books tag, Books do refer to how she got her knowledge of Westeros. From AGOT, Chapter 46:
My mother died giving me birth, and my father and my brother Rhaegar
even before that. I would never have known so much as their names if
Viserys had not been there to tell me. He was the only one left. The
only one. He is all I have
Also from AGOT Daenerys VI:
She had never seen the Seven Kingdoms either, no more than Drogo, yet she felt as though she knew them from all the tales her brother
had told her. Viserys had promised her a thousand times that he
would take her back one day, but he was dead now and his promises had
died with him.
Viserys had even told her about the Tourneys. From ADWD Daenerys V:
"One would be enough if he were as true as you." The day might come
soon when she would have need of every knight. "Will they joust for
me? I should like that." Viserys had told her stories of the
tourneys he had witnessed in the Seven Kingdoms, but Dany had never
seen a joust herself.
Also from AGOT, Daenerys I:
Yet sometimes Dany would picture the way it had been, so often had her brother told her the stories. The midnight flight to
Dragonstone, moonlight shimmering on the ship's black sails.
Viserys also told her about the Targaryen custom of incest. From AGOT, Daenerys I:
She had always assumed that she would wed Viserys when she came of
age. For centuries the Targaryens had married brother to sister, since
Aegon the Conqueror had taken his sisters to bride. The line must be
kept pure, Viserys had told her a thousand times; theirs was the
kingsblood, the golden blood of old Valyria, the blood of the
He also told her about History of House Targaryen's dragons. From AGOT, Daenerys III:
Viserys had told her that the last Targaryen dragons had died no more than a century and a half ago, during the reign of Aegon III,
who was called the Dragonbane.
Also from ADWD, Daenerys II, he told her about the Dance of the Dragons:
Viserys had told her all the tales when she was little. He loved to talk of dragons. She knew how Harrenhal had fallen. She knew about
the Field of Fire and the Dance of the Dragons. One of her forebears,
the third Aegon, had seen his own mother devoured by his uncle's
dragon. And there were songs beyond count of villages and kingdoms
that lived in dread of dragons till some brave dragonslayer rescued
It was also Viserys, whose knowledge had saved Daenerys' dragons from starvation in ACOK, Daenerys I:
Or rather, tried to feed them, for the dragons would not eat. They
would hiss and spit at each bloody morsel of horsemeat, steam rising
from their nostrils, yet they would not take the food . . . until
Dany recalled something Viserys had told her when they were
Only dragons and men eat cooked meat, he had said.
He also told her about the Throne room and Dragon skulls that adorned the walls. From ASOS, Daenerys I:
"Viserys talked of those skulls," said Dany. "The Usurper took
them down and hid them away. He could not bear them looking down on
him upon his stolen throne."
He also told her about Tragedy of Summerhall. From ASOS, Daenerys V:
Viserys had spoken of Rhaegar's birth only once. Perhaps the tale saddened him too much. "It was the shadow of Summerhall that haunted
him, was it not?"
He also told her about theology and faith. From ASOS, Daenerys VI:
Westeros had seven gods at least, though Viserys had told her that
some septons said the seven were only aspects of a single god, seven
facets of a single crystal.
So here we see, Daenerys knows all she knows thanks to Viserys. To sum it up, following are the subjects Viserys told her about:
- Her self, Her rights and Her House
- The tales of Seven Kingdoms
- Customs and culture of Seven Kingdoms
- History of House Targaryen
- Some mysteries of Dragonlore
- History of Dragons
- Faith of the Seven Kingdoms
So if this event does happen in the books, It would most likely be Viserys who would have told her about the pin and the office of Hand of the King.
There is no indication in the show that who told her about the office and its pin.
The exact dialogue in S06E10 was:
Daenerys: I, um... I had something made for you. I'm not sure if
it's right. Tyrion Lannister, I name you Hand of the Queen.
Since she wasn't sure about the exact appearance having never seen the pin personally but yet had it made, that implies she did so with help of verbal description she may have heard from someone. Following are the candidates who may have told her about the pin custom:
- Viserys Targaryen
- Jorah Mormont
- Barristan Selmy
- Tyrion Lannister
The most plausible one of them is Viserys since he was the one who told Daenerys everything she knew.