"Listen. Strange women lying in ponds distributing swords is no basis for a system of government. Supreme executive power derives from a mandate from the masses, not from some farcical aquatic ceremony. You can't expect to wield supreme executive power just 'cause some watery tart threw a sword at you. I mean, if I went 'round, saying I was an emperor, just because some moistened bint had lobbed a scimitar at me, they'd put me away!"
Although there are no indications that having the sceptre would make you ruler, just that the king has to keep the sceptre to keep the throne.
Not to mention that you only actually have to have the sceptre during a specific ceremony- it's actualyl a plot point that Tintin must recover the sceptre befopre the ceremony or the king would have to resign.
The Golden Helmet of Harold Hardrotta in the Disney Ducks Comic Universe confers the rulership of the entire continent of North America to whoever has it. It crops up at the worst times and in the worst hands. (Until a later comic had even more ancient pieces show up!)
Council Wars - Membership in the council is contingent on possessing a Key.
King Arthur - The original Sword in the Stone is introduced this way, and some adaptations run with it.
Note that the original legend actually subverts this; nobody cares when Arthur loses the sword and switches to Excalibur.
Although in that case, it had less to do with the sword itself, and more with the fact that Merlin said that the person who could take the sword would be the rightful king (which was just a flashy way of introducing Arthur, who was rightful king by birth).
Parodied in a footnote in one book where the person that found the magic sword consisting of two bits of wood nailed together is acclaimed king, because they also appear to have found several thousand heavily armed men along with it.
In The Queen's Thief, the MacGuffin is a stone carved with runes, said to have been dipped in the water of immortality. Whoever has it has claim to the throne of Eddis, which is exactly why Sounis wants Eugenides to steal it for him.
In Romance of the Three Kingdoms, Sun Jian finds the Imperial Seal in a well during the sacking of Luoyang. This gives him the "Mandate from Heaven" - in other words, the divine right to rule over China. However, he is killed shortly after leaving Luoyang, and the seal falls into the hands of Cao Cao.
In Queen Zixi of Ix by L. Frank Baum, the king dies without an heir, and the rules state that the crown goes to the forty-seventh person to walk through the city gates the next morning. Naturally, the story follows the ordinary boy who is the forty-seventh person that day.
In L. Sprague de Camp's Reluctant King series, the kingdom of Xylar chooses its next king by throwing the head of the previous king into a crowd—the catcher gets the throne. The catch is that in five years, the process is repeated... which is why Jorian, who had no idea about all this, is very much the titular Reluctant King, and spends the trilogy running away from Xylarians who want to drag him back so they can perform the ceremony.
The Founder of the Kingdom in the backstory of the Branion series appears to be this. She travels to the birthplace of her religion, meets her god, makes a pact with it, and becomes its avatar, a God in Human Form. Since it's a god of fire, this enables her to conquer the realm and become its first ruler.
In Patricia C. Wrede's Shadow Magic, the crown of Alkyra, a potent magical artifact, has been missing for hundreds of years. One of the characters finds it and puts it on in a moment of desperate need, and is somewhat dismayed to realize that this automatically makes her Queen. (The fact that what she needed the crown for was to defeat an invading army headed by demonic beings does incline other people to think this is a good idea.)
Babylon 5 - The Drazi factions are led by individuals with the leader sashes, making anyone wearing the sash the leader.
In the episode Coming Of The Shadows, a Centauri noble mistakenly plans to use a lost piece of crown jewelry to lay claim to the throne. Londo refuses part in it, explaining why the idea won't work.
There was some explanation, since the piece was a symbol of his own house more than of the Emperor. He is still heavily mocked by his allies for his stupid plan.
The legend of Odysseus / Ulysses ends on something of this note, with Queen Penelope announcing that whoever can thread her husband's legendary bow, which only he can use, and successfully use it to shoot an arrow through a dozen axe-heads will be named King of Ithaca. Of course, what no one knows is that Odysseus himself, who has finally managed to reach his home after a journey of ten years, is secretly one of the contestants...
In Exalted, one of the greatest and wealthiest nations of the South, the city-state of Paragon that has over a million people (not all of them in the city, some are just farmers) is ruled by the man who found an artifact of great power from the First Age. Additionally, there is a second part of that artifact that would allow whoever finds it to share that rule and impose his own will on the populace. Needless to say, it's still lost and it makes for a fine story hook/plot device/MacGuffin
Little King's Story begins with the player character finding a crown and putting it on, proving himself to be the destined ruler.
Hilariously used in Knights of the Old Republic. The Mandalorians had a special helmet that was the symbol of leadership. Revan permanently shattered them as a united force by beating the current leader and stealing the helmet (he gave it to a Mandolorian friend to rebuild them years later).
In How I Killed Your Master, whoever possesses the governor's imperial seal can issue his orders, making its bearer the effective ruler of the province. Subverted when Wong finds it by accident and asks if this makes him the governor. Fang Lin assures him that if he tries to use it, the factions hunting for it will just kill him and take it.