Note: Lint from a clothes dryer is bad, bad, bad. To see why, wet it and see how it mats up and takes forever to dry out.
If you have a pet such as a cat or dog that you brush, instead of disposing of the hair, put it out where the birds can grab it. For my chickadees, wrens, hummingbirds, ... I put hair in a suet feeder.
Picture from previous year of basic wire cage suet feeder.(foot#2)

Don't be surprised how quickly the hair disappears once the birds start nesting. And how much the birds can use. I have had two suet feeders stuffed with hair get emptied in three days after sitting untouched for several weeks.
Note: If you have recently put flea treatment on your pet's skin, I would wait a week before using hair from a brushing. I haven't been able to find any reference on suitable waiting periods, so this is simply my guess. Recognize that the birds--parents, eggs, nestlings--will have extended close contact with the hair in a space with poor circulation.
Many birders and others have noted that recently there has been a substantial decrease in songbirds locally, and there are multiple theories about the causes. Providing birds with materials to build better nests may help reverse this by increasing the survival rate of nestlings.
Stories:
I began doing this seriously in the mid-1990s when I was adopted by a cat who had thick fur and shed a lot. Weather permitting, I would brush him in my backyard. In the spring, I noticed I had a mix of birds 10-15 feet downwind competing for the fur: Some were snatching it out the air, while others waited for it to snag on a plant. One chickadee was so intent on not missing out that he kept trying to grab more even though his beak was already full and his birdhouse was only 10 feet away.
One of my neighbors had a Labrador-mix with white fur and after the nesting season, I let their children see one of the nests and they were amazed that it had a thick liner made from their dog's hair.
----Footnotes----
1. Whether there is a distinction between hair and fur is a matter of some debate, and it seems to be one mostly of usage. For example, humans have "hair" but not "fur", whereas animals have "fur" when it is on their body but "hair" when it is on your clothing and furniture. I debated having the title be "Pet Fur for...".
2. Suet feeders are typically $3-8 for the simple wire-cage version. Many Pet Supply stores carry them, as do various hardware stores, small and large (Orchard Supply, Home Depot)
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