So, I've come upon a possible scenario:
In 1078, the Varangian Guard revolted against the Emperor Nikephoros III after he blinded Nikephoros Bryennios, a general who had tried to declare himself Emperor. The Varangians had sought to kill the Emperor, but were suppressed by soldiers who remained loyal and had to beg pardon. So, lets say that Nikephoros is killed and the captain of the guard finds himself declared Emperor. This is a time of deep political conflict in the Empire with many factions, and its conceivable that many might prefer to see a neutral party assume the throne in order to keep their own enemies from taking it.
Now, there is going to be a series of revolts in any case, but assume that the new Nordic Emperor, lets call him Ragnar (since Emperor Ragnar I sounds kinda cool!), is successful in putting these revolts down. He also moves quickly, converting to Orthodox Christianity, and makes a strategic alliance with the Komnenos family and marries a member. Now, the Empire is facing a number of major conflicts during this time, including an invasion by Normans and continued conflict with the Turks. Assuming that Ragnar is successful is weathering the storm, and pushing these threats back, he might find himself as a popular Emperor and in a secure position.
One other possible thought: Rather than asking for a Crusade, I could see Ragnar turning his attention to central Anatolia, offering lands to any Nordic or Anglo-Saxon who come and help him drive the Turks from the region. This would draw a number people who would relish the chance to enrich themselves with land, and would also help create a seris of land owners that would be loyal to Ragnar and his descendants (of course, assuming, he is successful in his campaigns).
How does this work?