I wouldn't go so far as to say that the Sox have captured the latest generation of Chicago fans, but I think they've made a decent turnaround from how things were pre-2005.
I agree that the Sox lost probably two generation of baseball fans from the late 60s into the 1990s for reasons we've discussed umpteen times. In recent years, though, not only did the Sox win it all in 2005, but the Cubs have been plagued by a series of embarrassing fiascoes over that time: Bartman game, Sosa's fall from grace, playoff chokes in 07 and 08, 100-loss seasons, bad press about Wrigley's condition. Also, being on WGN TV is no longer fueling the Cubs' popularity like it did in the bad old days of the 1980s.
The Cubs still own this town, i'm well aware of that. But the Sox have won over a decent number of young fans. But, to continue that trend, the Sox need to become a team that regularly makes the playoffs. That's what is so maddening about the direction the team has taken after winning it all. Our last playoff appearance was 6 years ago. I'm encouraged that Rick Hahn and Co. have the team headed in the right direction again, but how long will that playoff drought be before the Sox make the postseason again? 8 years? 9? That's no way to overtake the Cubs in Chicago.
White Sox Baseball 2011-2016*
*except for 3 months in mid 2012 and 5 weeks in early 2016