Probably one of the biggest conspiracy theories in medieval times was the Jewish blood libel. Medieval Chirstians believed that the Jews needed the blood of Christian boys to make the matze, the Jewish unleavened bread they eat during Pessach (Jewish Passover), and that they captured, tortured and killed these boys. So whenever a Christian boy disappeared or was murdered the local Jewish community was one of the first to take blame. There are several accounts of Christian mobs killing Jews due to these rumors.
Some people in the Middle Ages even believed that the various Jewish communities in Europe met annually to decide which one of them would kill a Christian boy and share his blood with the others. Doesn't this look like a conspiracy theory to you?
Famous cases of Christian boys supposedly killed by Jews for the matze or other terrible rituals were:
Almost all these cases ended with a Christian mob killing or demanding the execution of many local Jews.
We know of these cases today due to a lot of this children being martirized by the Catholic Church of the time. Some of them have been removed afterwards (some only in the 20th century!).
The legend of Prester John may qualify.
Beginning in the 12th century CE, it continued to have influence over the European Christian imagination into the 16th century CE.
I was reading Over the Edge of the World: Magellan's Terrifying Circumnavigation of the Globe by Laurence Bergreen recently, and he mentions Prester John in the context of early explorers from Europe and their beliefs about the wider world.
Around page 35 of the Kindle version, he states...
"The legend (of Prester John) originated in 1165 when a lengthy letter began to circulate among various Christian leaders; as time passed, the letter became more elaborate as anonymous authors added beguiling, utterly fantastic details; so great was its appeal that it became one of the most widely circulated and discussed documents of the Middle Ages, translated into French, German, Russian, Hebrew, English among other languages, and with the introduction of movable type, it was reprinted in countless editions.
Addressed to Manuel, the Emperor of Constantinople and Frederick, the emperor of the Romans, the letter read, 'If you should wish to come here to our kingdom, we will place you in the highest and most exalted position in our household, and you may freely partake of all that we possess. Should you desire to return, you will go laden with treasures. If indeed you wish to know wherin consists our great power, then believe without doubting that I, Prester John, who reign supreme, exceed in riches, virtue, and power all creatures who dwell under heaven. Seventy Two kings pay tribute to me. I am a devout Christian and everywhere protect Christians of my empire, nourishing them with alms.'"
The author goes on to state that the legend of Prester John grew like Pinocchio's nose with each new addition. It was so widely read that Magellan's men were familiar with the legend of Prester John, and expected/hoped to find distant Christians beyond the boundaries of the known world.
So I believe this may qualify as a persistent late medieval /Renaissance / early modern belief that was widely held while being also completely false.