The phrase "day of birth" consists of two words in nominative and genitive cases, respectively. The second is a dependent word with respect to the first. Analogues: "Constitution Day", "Groundhog Day", "Day of Cosmonautics", "Day of the air force".
The second word is technically the indirect complement, and in fact – the definition describing the first. A day or what? – of birth.
The noun "birth" is used in the singular, in the genitive case. Refers to the average genus, is inclined to the second type, therefore, has the ending "I".
Verbal nouns, which include the word "birth" have two spellings: "ie" or "rd": a blink – moment, movement – the movement. So the options are: birth – birthdayi.e. In the phrase, respectively, will be used variants of "day of birth" and "day birthdayme."