He's a puppy and most likely just trying to play, but you have to make him understand that biting people is unacceptable. I personaly don't like the idea of a "time out" in a crate or kennel that is used for containment when potty training or leaving the house because they will get a negative association with it and you'll then have another problem.
A very firm "NO" and move him away form you (put him down off the couch for example). If he comes back up and does it again, repeat. He's finding your moving fingers much more fun than a toy, so after you've corrected him firmly and he is not doing it for at least a few minutes, get down on the floor with him and throw the toy around, see if he'll play fetch with you. He's probably bored and needs more play time, attention from you and stimulation.
The other thing to do if telling him no doesn't work is "bite him back" with your fingers (NOT literaly lol). Make a little "claw" shape with your hand and quickly touch him on the side of his neck (NOT his throat). Just make sure you do this in a matter of fact way so he doesn't think you're playing. If he was biting on his mommy and she didn't like it, that's what she would do to him, so he'll "get" that it means, this isn't allowed.
As far as your little cousins, most little dogs aren't naturally super fond of really young children. You have to make sure the kids don't get down on the ground face to face with him. I realize they don't understand (I have a 6 yr old son, so trust me, I get it) but I'm sure they understand following rules so you just have to make it completely understandable to them that it's a "rule" that they can't do that. Better yet, have their mom or dad explain it to them (I'm sure they don't want their children getting bit
). It's just as much about teaching the kids as it is the dog
When the kids are around (and not in his face), work to get him in a zone where he's focused on you and make him "sit" or "down" then reward him with praise or a treat. You basically have to teach him to be calm and not get nervous around children, but you have to be able to have his full attention first. If you don't feel like you do, I'd suggest getting help from a trainer or do a puppy class with him. Some will disagree with me , but I really don't like when people say "my dog doesn't like kids", I understand that, but I firmly believe it's important to socialize a dog to every experience possible and children are part of life.
You really have to nip that one in the bud quickly, or you're right, there's a possibility he'll become really aggressive towards them before long even though, most likely, he's just doing it in a playfull way now. I've taught my son how to treat animals and how to behave towards them so we don't have a problem here, but my 4 yr. old neice always wants to hold Alice and I just tell her, sorry honey, she doesn't want to be held right now, but maybe later when she wants to, I'll hold her and you can pet her. That makes my neice happy and Alice is learning that children can be nice to her too and there's nothing to be afraid of
sorry so long, but the children thing is a hot button topic for me