I think I have a fairly unique idea I would like to to share with you guys.
Harald Hardrada is known to have taken an expedition to Bjarmland(sp?). I think that his time at court of the Romans revealed to him the vast wealth of the eastern trade, and he was in fact looking for the north-east passage. Of course, his expedition failed, and supposedly was narrowly averted from total disaster.
What if Harald had a second attempt at the riches of the east though? This time sailing west towards Vinland. The fact that it would be impossible to reach Asia from said route isn't important. What I'm thinking is the investment by the Norwegian (or maybe even the English) crown towards finding a northwest passage to Asia.
What kind of resources would be available for an expedition? How many ships are doable? Is it likely that the Greenland colony is invested in? Would founding a colony in Vinland (or further south) to aid further exploration be likely?
For the moment let's ignore Hardrada's invasion of England. Either he wins or never attempts it in the first place.