As long as both dogs are well trained to understand that the humans are in charge, you shouldn't have any problems.
1) Prepare the home: The day you pick up the puppy, pick up all toys, bones, beds, and blankets before bringing your new puppy home. Put them away in a closet or cupboard where neither the lab nor the puppy will be able to see them. This will eliminate the risk of a dog fight over food, toys, etc. For about a week before getting the puppy, allow your dog to sleep on a towel for use in step 2.
2) Introduce the dogs: I suggest introducing the dogs on neutral territory, like a park that your lab has never been to. Start by walking your lab for 60 minutes around the area of the park. Do not take any toys, balls, or treats. Just walk at a fairly tiring pace for the dog. Once the dog is nice and worn out, take him to the park.
Meanwhile, during this hour, someone else should go pick up the puppy and take it home. Wash the puppy in the home, and dry it with a towel. Then, dry it a second time with the towel your dog has been sleeping on. This will remove any strange scents from the puppy, and scent it with familiar smells. This will help your dog accept it.
Then, take the puppy to the park. Begin walking with your lab, and have someone walk with the puppy right behind you. Gradually, slow down to let the puppy catch up with you. Continue to gradually slow down and allow the dogs to greet each other. They'll do some sniffing, etc. If there is a growl or a snap, immediately begin to walk the dogs again. Continue this process until there are no negative signs, then go home together!
For this step, I recommend your mom takes control of the Lab while meeting the new puppy, since he follows her.
3) Happily Ever After: Never allow the dogs to snap, growl, or fight over anything. Over the first 2 weeks, you can gradually reintroduce beds, bones, and toys to the house's common area. If something is fought over, you take it away and hide it out of sight IMMEDIATELY. Always supervise the dogs when they are together to make sure your lab isn't being overwhelmed by puppy energy. Make sure you take time each day to play with each dog separately, and make sure each dog has a place where the other dog is not allowed.
I suggest crate training your puppy for when you're gone, and when you're home, keeping his activity limited to the kitchen or other large, easy to clean room. Of course, you should allow the dogs to play together for perhaps 2 hours a day at first, more as they become used to one another.
Make sure to neuter your new puppy before he hits 6 months to avoid any hormonally driven conflicts, as well.
Your new puppy will not automatically look up to the lab if he has a dominant personality. More than likely, your puppy will be happy to follow the older dog until he hits adolescence. Either way, you have to teach the puppy to respect the lab, and the lab to tolerate the puppy. The ball's in your court on that one!