It's my understanding that the Blue Jackets, like all recent NHL expansion teams, were named by the NHL. There was an opinion poll, I believe, that may or may not have shown a fan "preference" for something, but the league made the decision and I don't know that any poll results were ever disclosed.
According to the team's website, the name was agreed to jointly be then-owner John McConnell and the league. A good guess is that Mr. Mac liked the name and the league approved. I know he was partial to "Blue..." and the team colors include blue, because Worthington Steel has blue in its logo. Mr. Mac founded the company.