With Tyson, it was the "holy left"

I found an article in the Budapest Sun that explains the mystery of Szent Istvan's right hand being preserved in the Budapest Basilica:

On regular days the Holy Right Hand (strangely, it is known in Hungarian as the Szent Jobb, or Holy Right) is on display in a dedicated chapel in the Szent István Bazilika (St Stephen's Cathedral), but on August 20 it goes on a procession. The procession of the Holy Right takes place around the cathedral, with the relic followed by dignitaries of state and church.

The history of the eerie bodypart is curious and remained unexplained for centuries. In 1951 one Dr Ádám Bockor examined the hand and offered an explanation.

He reached the conclusion that the 45 years between István's death and the opening of his sarcophagus in 1083 was enough for the complete disintegration of the corpse. The right hand was the highest bodypart of the corpse which was obviously lying on its back, and its preservation and mummification was because of the effect of the hot, dry air stuck between the rest of the disintegrated body and the cover of the sarcophagus.

Such a phenomenon is not rare in this climate: another example of it is the monastery of Brünn (today's Brno, in Czech Republic) where the mummified corpses of monks are one of the main attractions of the town. After István's death in 1038, turbulent decades followed with struggles for power, and Mercurius, the chaplain of Fehérvár (today's Székesfehérvár), the burial place of the saint-king, considered it safer to remove the corpse from it marble sarcophagus in the middle of Nagyboldogasszony cathedral and hide it in a tomb under the building. He removed the intact right hand and took it to a church on his own land on the banks of the River Berettyó to the north of what is now Nagyvárad (now Oradea in Romania).

Following István's canonization, King László visited the church, thus giving royal approval to the growing cult of the Holy Right Hand. The reverence of the relic became law enacted in the 1222 Aranybulla (Golden Bull), the Hungarian equivalent of England's Magna Charta.

The Holy Right then traveled to Fehérvár, Ragúza (Dubrovnik in modern day Croatia), and finally to Buda. The procession, started after the Revolution and War of Independence of 1848-49, resumed in 1989 after a 40 year forced hiatus under communism. While a military concert, trooping of the colors and changing of the guard in front of the Parliament, an air parade over the Danube and the procession are integral accessories of Saint Stephen's Day, the event that attracts the biggest crowds to both banks of the Danube as well as on balconies, rooms and roof tops overlooking the river, is the annual firework display.

So, there you have it. The right hand is kept in the Basilica only because the rest of the body had already disintigrated. Too bad, but honestly, would an entire body be as compelling as a right hand?

I didn't think so.