A lot of my friends' birthdays are in December
A lot of my friends' birthday is in December
You are talking about a set of friends and their birthdays. The set of birthdays is in December and their birthdaysare in December. The second of your examples is incorrect; saying "A lot" does not make their birthdays a unit in the same way as referring to them as a set or a group would do ("A lot" here is synonymous with "many").