I've got a 8 week old mini daschund whos' eyes are watering off and on. Also, it looks like the hair around the eye is beginning to fall out. It's not really noticeable, but my wife is concerned.

What could be the problem, or do I need to worry about it at all.

Whatever the problem is, it needs the attention of a vet. I would make an appointment. Take any medical records you have for it.

For more on caring for young puppies, see the sticky at https://www.askmehelpdesk.com/dogs/information-articles-our-dogs-expert-labman-53153.html#post251802