Not sure if I have any good suggestions, but I also recently had a weird thing happen with my lips. It was either an allergic reaction to some cream I put on the outside of my lips (because they were feeling especially dry) or if I might have gotten a bacterial infection (staph). My baby, who has had eczema for about 3 mos. now, has a bacterial infection on his cheeks and I purposely don't kiss his cheeks, but will kiss the sides of his face. Well sometimes he moves and when he does I sometimes get his weepiness on my lip area. I think this may have been what happened. The pediatrician said that could be. She told me to put a little neosporin on. I do think it helped overall, but I haven't used it for a few days. I do tend to use chapstick quite a bit also, but have been trying not to. My lips are doing much better, practically healed. Still not sure what it was exactly, but I thought I would share my experience. It all seemed to start with a dry, itchy, burning feeling. I knew something wasn't right when I saw what looked like contact dermatitis (all red...like a burn) along the edge of my upper lip.
Another thing to think about is that I read about 30% of people with eczema have an allergy or sensitivity to neomycin (an ingredient in neosporin) and I think a lady on the website (eczema.net?) said that she had a reaction to the neosporin a week later. So that's something to keep in mind. (THAT... and a possible bacterial infection?)
Hoping this might help in some way. God bless!