Any time the Big Ten mentions expanding to twelve teams Notre Dame is always one of the first schools to come up. Also, every time the expansion question arises, Notre Dame rejects the Big Ten's offer. While a steep hill of tradition and money is preventing, and likely will continue to prevent, Notre Dame from joining the conference, there are several on the field reasons for them to do so. They all boil down to giving the Irish something to play for and some competition to back up a winning record.
First off, Notre Dame has very little to play for right now. For them, it is National Championship or bust. Yes, the BCS Bowls are nice consolation prizes, but Charlie Weis made two of them and look where he is now.
Joining the Big Ten would give Notre Dame a smaller goal to judge their success. Instead of always shooting for the biggest stage, they can hang their hats on a Big Ten championship and a BCS berth. Look at Michigan, they are one of the most successful programs in the history of college football and they went nearly 50 years between National Titles. They kept their reputation by winning the Big Ten 42 times.
It would also take pressure off of a coach. Notre Dame has fired coaches because they fail to get them to the ultimate stage. Winning a National Championship in the FBS is no easy task. You need to compete with 119 other schools, win all your games, and get the media to like you enough to vote you as number one. Winning a twelve team conference is much easier than all that. Giving the team a smaller, arguably easier goal will keep the coach around long enough to build winner.
Joining the Big Ten would bring Notre Dame better and more consistent competition. For years Notre Dame has caught flak for having an easy schedule. Sure, they play Michigan and USC every season, but who else do they play? Navy, Michigan State, Purdue and any school that wants to take time out from their more important conference schedule to give Notre Dame a shot.
Joining a conference would give them a constant and competitive schedule every season to back up their wins. Instead of playing Washington State or Connecticut they can play Wisconsin and Iowa. These are teams that look better on a BCS resume. Yes, they would have to play Ohio State and Penn State in most seasons, but isn't playing good teams better for making a National Champion?
Most importantly, to satisfy the traditionalists out there. They could keep every one of their annual matchups. They will play Michigan, Michigan State, and Purdue as a part of the conference schedule. With four out of conference games to schedule they can use them on Navy, USC, and Stanford and then one other wild card school if they wanted to.
Joining the Big Ten would allow Notre Dame to play for more than just a national championship and give them a tougher schedule, all while allowing them to keep the traditional rivalries with USC and Navy alive. Sadly, a lucrative television deal with NBC and a bunch of traditional fans and university executives will keep Notre Dame an independent team, forcing the Big Ten to look elsewhere for its new member.