My cousin wants a PS4 for his birthday but he doesn't have Internet in his home. My friend said Internet is needed, but I heard that it's not. If my cousin only wants to play games locally, will he still need Internet?
It depends only on the game your cousin is going to play. A lot of games (e.g. Destiny or the upcoming Need for Speed) today work only if the playstation is connected to the Internet. But overall internet is not generally required.
What you can do
What You Can Do We tested the PS4 with multiple disc-based games offline and unpatched, and while most titles installed their own utterly mysterious software updates from the disc before running, they all worked. We tried Battlefield 4, Madden 25, Knack, Killzone: Shadow Fall, and Injustice: Gods Among Us Ultimate Edition, and while the multiplayer options of those games were obviously disabled we could play the single-player modes just fine. Of course, this only applies to disc-based games unless you patched your PS4 and downloaded other games to the system already.
What you can't do
Unpatched and offline, you can't set up PlayStation Vita features like Remote Play or Second Screen, and you can't set up facial recognition with the PlayStation Camera to log in automatically (but you can set up multiple users for couch-bound multiplayer). You can't play online games, obviously, or access online services. You also can't play downloaded games when not signed into PSN, so you're bound to disc-based titles. The PlayStation Dynamic Menu loses the majority of its information offline, with no activity feeds or suggestions, and the "What's New" menu the PS4 defaults to on startup will look very bare. And, of course, you can't use any social media features. If you can't get online or if you can't download the PS4 pass at launch, good news: you can still play (disc-based) games and you can still capture video and screenshots. You just can't really do much else.