It might sound overly simplistic, but I feel that there are a few factors that really decide whether or not a given team is "built for Omaha."
The first is depth in the pitching staff, particularly in the bullpen. If a team is going to advance to the championship round of the College World Series, they will have to play a game basically every other day for a week. That can take a toll on a pitching staff.
The teams that really excel are the ones that have several different bullpen pitchers they can go to in case a starter has a short start. If you don't have that, you can find yourself throwing the same two or three guys all week and running out of pitching pretty quick.
The next is having an offense that isn't overly reliant on one player. Having a transcendent offensive player is great. There are surely times when that one guy can carry your entire offense. But when you get to this stage of the postseason, you are going to be facing some elite pitching staffs. Those elite staffs will be able to shut down that one player and in turn, your entire offense if he is the only guy that can beat them. Getting contributions all up and down your lineup keeps teams from being able to work around your star.
The third thing is having a shut-down closer. It sounds like a cliche, but the last three outs of a baseball game are the toughest to get. This is particularly true if you don't have a closer that can come in and shut the door against anyone.
With a good closer, you can turn it into an eight or even seven inning game and rest easy knowing that you have a guy that can finish it off.
Out of the 16 remaining teams in the tournament, there are a few that really stand out as teams that are made for Omaha. Let's single them out and examine what it is that makes them so likely to succeed in the College World Series.