I've been toying around with the idea that along with the Teutonics, the Hospitallers are invited to help fight the pagans of the Baltic Sea.
They would need be to be patroned by Imperials, which wouldn't be that hard, but their main legitimacy base was about Holy Land and Mediterranean protection IOTL, with strong links with maritime cities and Yerosolemite Kingdom. Switching this wouldn't be done without a small crisis IMO.
They fight along with the other Northern Crusaders and end up carving out a state in southern Sweden or Finland.
Southern Sweden was mostly, politically speaking at least, Christianized at this point, and with a strong focus on Finland. Safe big changes, it's unlikely Sweden would patronize Hospitaleers as HRE did with Teutonic Order or their predecessors as in Riga.
If not, what about the Order taking over somewhere in Italy?
Taking over, as a conquest? Probably not. Having holdings in Italy, as they did in France? Well they already have so, and I suppose they could grab more with a more critical Mediterranean situation.
Maybe a crusade on Genoa after a Genoese alliance with the Ottomans?
Unlikely : Hospitaleers and Genoese had important links with each other, and the maritime city would have few interest forging an alliance with Ottomans, if at all.
A crusade against a Muslim Sardinia?
Sardinia, when religious knights orders were established, were clearly outside Muslim influence since decades.
Another EDIT: What about Normandy? The Vikings were pagan when it was established.
Most of Scandinavian settlers were actually Anglo-Scandinavians, which means at least partially Christianized. Note that Rollo and his men went Christian road as a condition of their establishment, making a mix of Frankish and Scandinavian population realized after one, maybe two generations at latest.
By the end of the XIth century, Normans were entierly part of the French cultural continuum, including religiously.
not to mention that the Norman invasion of Britain was approved by the Pope as a holy war.
It probably wasn't the case, and added later.
My favorite PoD would be that the parlament in one of the iberian states (cant remember which) honours its Kings testament that left the country to the Knights of St John and the Templars
Alfonso the Battler tried to do so, but it meet a large resistence : he simply couldn't have gave away lands that he inherited (at best, the lands he personally conquered).
It may even be possible that issuing an half-absurd will was his original attempt, in order to fend off foreign influences on Aragon's succession.
For mere matters of legitimacy, and maybe legality, the pareage inheritence by knights orders wouldn't pass; and the fierce opposition of Navarra and Aragon's nobility would make it extremely hard. (We're not talking of Parliments there : they simply didn't existed)