Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Birmingham, Alabama to Bristol as you normally would.
In terms of precipitation, rainfall in Bristol ranges from 2.0 - 3.7 inches per month. November is typically the wettest month, when rainfall can reach 3.7 in. May is typically the driest time to visit Bristol when rainfall is around 2.0 in.
If weather is an important factor for your trip to Bristol, use this chart to help with planning. For those seeking warmer temperatures, July is the ideal time of year to visit, when temperatures reach an average of 64.4 F. Travelers looking to avoid the cold should look outside of December, when temperatures are typically at their lowest (around 39.2 F).