The Giants, might have a bigger rivalry with the Chicago Bears and San Francisco 49ers over the years, if you look at all the great games they've had with both franchises. Especially the 49ers which goes back to 1981 and where they seemed to be playing each other on Monday Night Football every year in the 1980s, or in the playoffs. As you see in this video most of the great games that the Giants and Cowboys have played against each other has been in the last 5-10 years or so. So you might be able to say that the Giants-Cowboys rivalry has been great during this period, or is one of the best rivalries in the NFL right now. Similar to the Ravens and Pittsburgh Steelers, or the Ravens and Cincinnati Bengals. But historically the Giants main rivals have been the Eagles and Redskins and then after that the 49ers and Bears, as far as the big games that they've played. With the Cowboys it has been the Redskins and Eagles and then the 49ers as well.
To me at least great NFL rivalries are historic. They survive through the years with people being able to say, "remember that great game against them thirty years ago when we beat them for the division? Reminds me of that game last year when they beat us for the division." Just because a rivalry is hot, or sexy, or however you want to phrase it, doesn't make it a great rivalry if no one cares about it 3-5 years from now. Because both teams have moved on, or perhaps are struggling now, or just one team is struggling with other getting the better of their games. And yes since both the Cowboys and Giants have been regular playoff contenders the last ten years or so and generally have both been in the NFC East race, they've played a lot of big games against each other and have done that in prime-time. But that doesn't put it in the same class as the Redskins-Cowboys, Giants-Eagles, Redskins-Giants, who go back forever against each other and have played a lot of great games against each other going back fifty-years. The Cowboys first big rival was the Redskins. Then the Eagles and then the Giants.