Not an expert, but I do not think that you need an evolutionary explanation: birds are often not stupid, and can simply observe that nests with more cigarettes have less parasites.
Do birds spend time observing the nest of their peers? AFAIK the only nests they care about are their own (though I'm definitely not an ornithologist, I'm mostly ignorant about them in general).
But that's the beauty of Darwinism, whether the information is stored in the genes or in their culture (memetically) doesn't matter, the selection/amplification process is the same.
We know they're capable of passing complex information onto each other. What "a bad man's face" looks like for instance, I'm sure some of HN has seen the documentaries on their abilities.
I wonder if they choose nesting material based on insulation properties also. Cigarette stubs are probably great for stuffing into cracks and it is known that plugging air gaps dramatically increases whole house/nest insulation value.
But that's the beauty of Darwinism, whether the information is stored in the genes or in their culture (memetically) doesn't matter, the selection/amplification process is the same.