The sentence you wrote in the end, "The father of one of my friends is serving in the Navy", conveys the meaning well.
If you write, "one of my friend's father...", it means you are talking about a particular friend (since you are using friend's, singular). And you are referring to one of his (the friend's) father. Which is the same as saying, "one of the child didn't come" as opposed to the correct usage, "one of the children didn't come". So, you should change father to fathers, if your friend has got more than one father. If not so, and you leave the sentence as it is, it's incorrect.
In, "one of my friends' father...", you are talking about one among many of your friends (since you are using friends', plural). And you are referring to that friend's father. The meaning of this sentence is the same as that of your last sentence.
Note: It's always "one of many"; not "one of a single entity". "One of" has to be followed by a plural.