The more standard story is that the bad weather made navigation almost impossible and people were getting sick, so they had to stop somewhere. They began searching for a suitable place to settle. However, it was the fact that they were running out of provisions--especially beer--that finally made them realize they couldn't be picky. And as anyone who's lived in New England can tell you, the only way you can voluntarily choose to settle there is if you're not picky about climate, or really anything else.
"We could not take time for further search or consideration,
our victuals being much spent, especially our beer" -The First Goddamn Americans (1622)
Since beer was boiled, it was cleaner than what passed for regular water at the time and therefore safer to drink (especially during long trips). So, if the beer in the Mayflower hadn't been running low at the time they were passing near Plymouth, they almost definitely would have continued their trip further down south, as close to the Hudson River as they could make it.