although it's written like a one word, havalimanı comes from the compound noun hava limanı. liman is port and it's literally airport. the -ı is a possessive ending.
after the possessive -(s)I the noun cases change a bit. all of them basically have the letter n in their beginnings so they become as below
-nI, -nE, -ndE, -ndEn
I think it's the compound noun that confuses me so much. I'm trying to apply this rule (from possessives section): "The objects being owned have the ending -(s)I. The owners have the ending -(n)In", but the problem is that 'hava' has no ending, so is it usually so that the 1st part of the compound noun gets no possessive suffix?
if the first part (owner) takes an article like a/an or the in english it usually takes the genitive ending -(n)In
the card of a student - bir öğrencinin kartı
a student's card - bir öğrencinin kartı
but freedom of speech - ifade özgürlüğü. because speech has no article, it's a general term
a student card - bir öğrenci kartı
an airport - bir havalimanı
a telephone number - bir telefon numarası
note that in these examples the owner doesn't have any article. the whole compound noun has article