1. Blue helps with headaches.
(What is the meaning of 'with' in this sentence?)
I don't understand what you mean by this sentence.
If your sentence were: "Aspirin helps with headaches," then it would make more sense.
The preposition "with" in this sentence means "in some particular relation to (esp. implying interaction, company, association, conjunction, or connection): I dealt with the problem. She agreed with me."
From Dictionary.com