That is unfortunate - especially given your daughter asked the person not to. Did your neighbor pet the dog on it's head? Not trying to blame the neighbor (tho arguably they didn't heed your daughter telling them not to pet) but this is sometimes seen as a dominance move by dogs. Some dogs also just don't like men, men with hats or people whose eyes they cannot discern. Not trying to remove the responsibility you have as the owner of the dog, but it is good to keep a mental profile of things which may trigger your dog's aggressive behavior.
This can be a teaching moment for you and your family. For the immediate future, presume that your dog will bite again and take measures to prevent this. Harry V. makes an excellent suggestion about the basket muzzle and regular training. In general a regular schedule with your dog will do a lot to keep their temperament even, e.g. feeding, doing bathroom business, training, play time, sleep and down time. Play it safe until you establish a routine and the dog knows who its master is and where its place is. It is good to be safe, but I have never personally seen pit-mixes get aggressive towards their own family members. So, I wouldn't worry, but be cautious.
If you are not already familiar with them, look into your local regulations regarding dog bites. In some States, "bully breed" laws allow for automatic rulings in cases of bites. I don't say this to encourage you to give up on your pet, just so that you know the liabilities. I have owned pit mixes without any incident of biting and do find them to be excellent dogs, however, knowing how they are perceived and the laws regarding their aggression, I am cautious with them around others, in new environments and when meeting strangers. Knowing the liabilities, the reward of owning such a dog is loyalty, love and an otherwise wonderful pet. I have found pit-mixes to be excellent pets - very curious, attentive, energetic and friendly.
Is your daughter strong enough to restrain your dog? Even smaller (~40lbs) pit mixes can be surprisingly strong. It may also help to get your dog a harness for on-leash time so that if they need to be physically restrained, you are not risking damaging their throat by pulling too hard on their leash.
Do you have a dog park nearby? In addition to regular training in the home and neighborhood, socializing your dog with other dogs will help their behavior. Take it slow tho. As with any new environment, do not force them into a new situation. This might overwhelm them. I find it is a good practice to walk the perimeter with the dog while on-leash and let them get to know the area prior to letting them off-leash inside a dog park. The walk will help acclimate them, and give you an indication of how they will relate to the other dogs there. Start with short visits and keep a very watchful eye on them.
The main thing, tho, is that you need to train your dog so that they know their place and respond to commands. If they do not respond to basic commands of "sit", "stop" or "off", "lie down" and such, then do not let them off the leash. Once off the leash, they must also continue to respond to commands. It may help to use a stern command voice. Once the command is followed, repeat the command in an affectionate voice and praise them, e.g. sternly say "sit" (it might help to raise your hand while saying this: eyes up, butt down). After they sit, smile and say, "good sit." Then sternly say "stay." At that point you should be able to turn away from them and ignore them and they stay in their spot. When you return to them, smile and say "good stay."
Normally I suggest training without a lot of treats to reward good behavior - instead using your voice and affectionate touch. In the case of a dog that is already a year old, treats may be appropriate, just don't over feed them. Also, be wary of them training you to get the treats they want ;)
Be very strict about not letting strangers pet the dog. If they insist, tell them to not pet the dog on their head. It's a better technique to first tell the dog to sit. Then you have their attention. Then introduce them to the new person. You might also try the "paw" command so that your dog will hold up their paw to shake hands.
Lastly, speak with your vet about behavior training.
I hope things work out well with your family and your pet.