At what point did it become Robert's Rebellion when Jon Arryn was the one who initiated the rebellion?

Because the rebellion began with Robert Baratheon's ire. The pot reached its boiling point when the dragon prince abducted the wolf princess. It ignited a chain of events that led to the death of Eddard Stark's father, brother, and, eventually, including his sister. Stark, Baratheon, and Tully men summoned as much people as they could. When all the men had decided to band together and stand against the unjust Targaryen, they stood behind Robert Baratheon as he was a Targaryen grandchild, so he had the best chances to the Iron Throne. Ergo it is why the event is aptly named the War of the Usurper.

The name is given by the people, the singers, the common folks. People rally to a charismatic, powerful rulers. Singers will sing about the victories won on battlefields. Rarely will you see glorious songs written about strategists or wise old men who advise the protagonist. In the case of this rebellion, though Jon Arryn initiated it, Robert was the face of it. He was the warrior who led the armies into battle. He was the one who won at the trident. Hence his name is on the rebellion. To answer your question more specifically, I think it was coined Robert's Rebellion the moment it became big enough for people to start referencing it with a name. Another possibility is the cause for the rebellion. Robert's cause is the one that is easily understandable by the people and also makes for a good song or a story. A hero denied the love of his life and goes against the perpetrator for revenge. Where as, Jon Arryn, does it out of ideals or out of political ambitions (if you believe in the Southron ambitions conspiracy) or to depose the mad king. Now which do you think fits better into a song? It could have been Ned's Rebellion though but again I think when it comes to charisma and military leadership etc... I think Robert would still win. I am not sure of Stark's popularity amongst the common folks in south. Also the fact that Robert killed Rhaegar and ultimately sat on the throne should have sealed it.

The Baratheons have a minor claim to the throne as they are descended from the Targaryens on their mother's side, so I think they might have made him the face of the rebellion for this reason as well.

Jon Arryn did not command Robert Baratheon or Eddard Stark. Jon was not liege to Robert or Ned. They were lords of their own lands, of equal standing. Usually wars are named after the fact, after the outcome is known. So the conclusion of this war was when King Robert ascended the throne, so it's called Robert's Rebellion by supporters and War of the Usurper by Targaryen sympathizers. That only makes sense after a usurper takes the throne. It's a martial society where truth and Gods' grace is determined through combat. While Robert fought many battles as he hammered through and up to the Trident, both Ned and Jon were securing those lands to the north through politics. Arranged marriages to the Tullys. One can very well argue that the efforts of Stark and Arryn to bring the Riverlands to bear on their side was what turned the tide, but singers don't make songs about the brave and grand matching of Jon and Lysa on the field of the marriage bed. They sing of Robert and the Bells. Of Robert and Rhaegar. Not to say a song about Jon Arryn and Lysa Tully wouldn't stir the heart, the brave man Jon was. King Robert won the throne by right of conquest. By right of his blood ties to the Targaryens. So the victory was his.

There were four major families who led the rebellion - Stark, Baratheon, Tully, and Arryn. From these families, Stark has the biggest cause for this rebellion as it was a Stark girl who was abducted and Stark family head and his heir killed by the Mad King. So technically it should have been Eddard's Rebellion but since Ned had no interest in becoming a king, and after Robert was crowned it seems fitting to call the rebellion after the king's name in songs, books, and folklore. Robert also had many other points in his favour. He was one of the affected party as he was to marry Lyanna and thus wanted revenge on Targaryen. He was to be killed on Mad King's orders when Jon Arryn defied Mad King and raised his banners. He was leading battles as a commander and winning them, his reputation as a fearsome warrior was widely known and he killed the Prince of Dragonstone. All brownie points for which you can easily get a rebellion named after yourself.

It is referred to as Robert's rebellion because Robert won; he killed Rhaegar, Crown Prince at the time and after Aerys II's death, Robert ascended the Iron Throne. Another reason is that the formal declarations of war that the original three Houses used when declaring war against the Targaeryens. House Stark rebelled to avenge Lyanna's abduction and the murders of Lord Rickard and Brandon, his eldest son and heir. House Arryn rebelled because Aerys commanded Lord Arryn to hand over Robert Baratheon and Eddard Stark, Lords of Storm's End and Winterfell, to the Iron Throne to be executed as one was Lyanna's brother and the other was betrothed to her. House Baratheon however rebelled for a different reason, at least formally. Robert condemned Aerys for his actions and contended that Aerys's crimes had tainted his entire bloodline, which was descended from Aegon V through the male lineage. He also said that Rhaegar and Viserys were unfit to rule Westeros because of this taint. Then, Robert pressed his own claim to the Iron Throne as a descendant of Aegon V through the female lineage, as he is descended from Rhaelle, Aegon V's daughter.

In westeros,politics and customs rule A lord has a responsibility towards the peasants that live within their land as such, all lord answer to lord above them and so on,with the king being the highest power. The seeds of rebellion was sown for long time,as  implied, no one dared stand up or revolt against the targ dynasty cause of fire breathing dragons The reason robert is seen as the figurehead is of course the fact that, no one is gonna side with a false claim to the throne,as the rebellion intended to overthrow the mad king and place robert on the throne (The baratheons can claim the throne because of blood from his mother side, and  also the house of baratheon  was established by Orys Baratheon , who was a close friend of Aegon the I and was also rumored to be a bastard brother of his) Hence the baratheons always had a close to the Targs and So with this in place, with arryn calling the banners and this claim, the rebellion can gather strength with robert working as propaganda and with jon pulling the strings behind the scenes.