It's a little bit complicated but here goes:
The Xbox 360 used to have some major hardware problems that Microsoft had very hard time fixing. If I remember correctly, their hardware failure rate was upwards of 50% which is totally unacceptable. That was years ago however, and the new Xbox One system is much better, and while the percent hasn't been publicly released yet, it is thought to be somewhere under 5%. That, coupled with warranties means the system should work fine for a long time to come.
1) $300 with a free game (which also costs $60 ususally) and a $60 gift card is a pretty great deal. You're basically being given $120 for buying the system which is impressive.
2) Xbox One Backwards Compatibility was a big gaming news headline when it was announced at this year's E3 Press Conference. The Xbox 360 games that will work on the Xbox One can be found here:
http://www.xbox.com/en-US/xbox-one/backward-compatibility/available-games
You can google "Xbox One Backwards Compatibility" for more information, but the gist of it is that yes, decent number of 360 games will work, though you may need an internet connection to get them due to how the compatibility works.
3) It is also important to remember that the Xbox One has already been out a few years. Soon, game publishers will stop spending money making versions of their games for the Xbox 360. So the new, often called "Triple A" games will not be on that system soon, and already some of this years biggest title's won't be on that system.
2015 has been one of the biggest and best years in video games there ever has been. I would say getting your son an Xbox One is a great idea. If he likes shooting games, he may enjoy Destiny or Call of Duty Black Ops 3. I would also highly recommend Fallout 4, as it is one of the best games I've ever played.
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Edited to add: I (narnia_girl) know very little about gaming systems, but my son is in college earning a bachelor's degree in video game design. I asked him to give a response here.