You’ve expressed before that Torrhen’s Square would be the best place for the North to raise a western navy and made some pretty compelling arguments. My question, could Bear Island be a good place to raise such a navy? Bear Island has timber for ships, fishermen who might be trained to become sailors, and the Ironborn have used it as a base for naval attacks before so it can be done. Assuming a good initial investment (from a rich dowry maybe?) couldn’t this work? Or am I missing something?

You make an excellent point.

There is one drawback to Bear Island tho: it’s really far to the north, which means response times against threats from the south are going to be really bad. And since the major naval threat to the Starks is the Ironborn to the south, it’s less than ideal…for a primary naval base.

As a secondary naval base, it’s excellent. All the advantages you speak of are absolutely relevant here: 

• First and foremost, if Torrhen’s Square can just delay the enemy, Bear Island can provide crucial reinforcements, allowing the Starks to weather initial attacks and then decisively counter-attack. 
• Second, because of Bear Island’s resources, ships from Torrhen’s Square that get damaged in a fight can quickly pull back to Bear Island, repair, refit, replace any dead crew, and then get back in the fight, making the Stark navy much more resilient. 
• Third, it also would help to deter seaborn Wildling raiding parties.