This is commonly from an upper respiratory infection, which is from several viruses (like herpesvirus, etc.). The good thing is none of them are contagious to you. Lysine is certainly benign enough. Make sure to isolate your cat from other cats (i.e., if she's an outdoor cat, she may be catching this outside from other cats). If she's eating well, and doesn't have any discharge from her eyes or nose, then it likely will come and go (the way a cold sore comes and goes when we're stressed!). If however, she isn't eating, has GREEN discharge from her eyes or nose (especially her nose!), is squinting her eyes a lot, etc., then she should be seen by a vet to make sure she doesn't have a secondary bacterial infection on top of a viral infection. We don't treat VIRAL infections with antibiotics - only if they are complicated by bacterial infections (green pus!).
Hope that helps!
Lysine will "starve" many viruses, but not kill them and will do nothing for a bacterial infection. ie it may be beneficial and won't hurt, but it is likely not enough. She should see a veterinarian if at all possible.