This question was asked over the internet recently. Not specifically to me, but it looked like a fun exercise. So, I took a look at answering the question with two different methods.
Dustin Byfuglien has played as a defender for this franchise for four years (well... one of them was part forward, part defenseman). In that time he has out shot his opponents 2686 to 2436 for 5v5 situations, giving him an impressive shot differential of +250. While Byfuglien's shot differential is very positive, his goal differential has been negative over the same four seasons.
Method 1: No Regression
Let's ignore that the Bruins are not just a better team in the net, but also in their shot differentials. The easy way out then would be to to take Dustin Byfuglien's shots against and use Boston's average on-ice save percentage. But, then we would have some Jets' fans whining that Byfuglien deserves his on-ice save percentage. Maybe they would mention things like shot quality, or not all shots are equal.
Okay then... To appease that crowd, let's look as if Byfuglien's effect on save percentage is 100% on him. We will call it the "Byfuglien Effect".
For 2010-14, Byfuglien has seen his goaltenders' save 90.64% of the 5v5 shots thrown at them. For that same timeframe the Jets/Thrashers have averaged 91.70%. This means that Byfuglien has experienced (in this scenario deservedly so) a relative -0.0106 on-ice save percentage.
(For number nerds, this is not an on / off ice relative number... for that see below)
The Boston Bruins however have averaged a 5v5 0.9345 for on-ice save percentage over the last four seasons. We add on the "Byfuglien effect" and get an "expected" on-ice save percentage of 0.9242.
Change Byfuglien's on-ice save percentage to 0.9242 and we get a Dustin Byfuglien with a 5v5 goal differential of +24 over the past four seasons.
Method 2: Regression
The problem with the last method is we know that this likely grossly over estimates the "Byfuglien Effect". Goals are extremely rare and the slightest luck or bounce can throw a goal differential way off course, even at multi-season samples. For example, Byfuglien -a ""terrible"" defenseman on a team that has constantly been outscored- only has a -19 goal differential for 5v5 over 4 seasons combined. That's it.
It's been shown that an 85% regression is typical for a three season sample for on-ice save percentage. This translates to on average 85% of that "Byfuglien Effect" is likely not because of Byfuglien, even if he is legitimately causing his goaltenders to save less shots per shot against. Now, I'm using a four year sample, which likely regresses slightly less than 85%. However, since method 1 over estimated the "Byfuglien effect" we might as well under estimate too.
The regression reduces the Byfuglien effect from -0.0106 to -0.00159. This shifts Byfuglien's 5v5 goal differential from +24 to +46!
So, there you have it. Byfuglien, even when saying he deservedly hurts his goaltenders save percentage, does pretty good job when moving from bottom 5 goaltending tandem to top 5. His goal differential over a 4 year span shifts from -19 to anywhere between +24 and +46. In addition, the true value is likely much closer to the +46 than the +24.
This is an issue that trips up people. Why does defensive play have such low impact on goaltending? Shouldn't bad defensive play mean more quality shots against?
The reason is that more quality shots against are not the only thing poor defensive teams allow; they also allow more non-quality shots against. The ratio may budge slightly one way or the other, but over the span of a season (or even a few seasons) there are multiple factors that may have a larger influence. Factors like bounces, luck, goaltending, line matching, line mates, etc.
It's not that all shots should be viewed as equal. Not at all. A shot in the scoring chance area has nearly a 14% chance of scoring, while all other shots are about 3%. The issue is that hockey is a fluid and highly variable sport. A lot goes on and players are not able to sustainably control the other 9 players. In the end, players at the extremes tend to regress heavily towards the mean.
Additional and topical note:
This article is not a shot at Ondrej Pavelec. Yes, he was the major cause of the poor on-ice save percentage behind Byfuglien (Chris Mason and Al Montoya's first season didn't make things any better either). However, he has apparently come to camp though in the best shape of his life, after years of having his conditioning questioned ever since his Draft+1 season. I'm still highly skeptical it amounts to much, but as a Jets fan all I can do is hope it does. Yes, I hope for Pavelec to succeed, even if I believe it to be unlikely.
Post-Publish Extra, Third Method: Relative On/Off-ice Values
I didn't use on / off-ice numbers for calculating the "Byfuglien Effect" but some were asking about it. As you requested, here is the same process but using the Jets/Thrashers' on-ice save percentage without Byfuglien on the ice:
Without regression we get -0.0152 instead of -0.0106, which translates to a +13 goal differential instead of +24.
With regression we get -0.00228 instead of -0.00159, which translates to a +45 goal differential instead of a +46.
Again, we know that Byfuglien's true value lies closer to the regression than the one without.