I am writing about my local airport which has had two other names which name should I use through out the writing about it? An example is JFK in New York was originally called Idlewild Airport would most people writing use John F Kennedy (JFK) through out their story?
When you're writing, the goal is to say something. Preferably, more than one something:
When discussing a real-world place, you (usually) want the reader to know what place you're talking about. If your primary concern is to communicate something about the place, obviously you should use the name readers are most likely to be familiar with.
When a character in your story is describing a place (or anything or anyone else), how that character chooses to do the describing says something about that character. If your character refers to JFK Airport as "Idlewild", that tells you something about the character - maybe that the character has not changed to get with the times, or perhaps that there is another (minor) JFK Airport somewhere that might be conflated... And if the character always refers to it as "John F. Kennedy Airport", without abbreviation, that says something else again.
When a narrator talks about a place in a particular way, it can say something about the story. If the narrator persistently refers to the land which JFK Airport happens to be built on by some fantasy name, this can signal what kind of story is being told - like an urban fantasy where locations have ancient magical significance. On the other hand, a story denouncing European colonization might try to use old Native American names for places in or around New York.
Whatever name or word choice you make can communicate one or more things to the reader, depending on what name you choose, as well as on "who" is saying whatever is being said.
Without knowing your intentions, your characters, and the location and history of the location, it's impossible to give a specific recommendation. But you can examine your possible choices through all of the above-discussed lenses.
To go along with Jedediah's comment, the content of your writing is also important.
If your subject relates to the history of JFK Airport, I would first inform the reader about the connection between the airport's older name and the new name. I would continue to use "Idlewood" as the name of the airport if the subject takes place before the airport was transitioned to its new name.
There are 3 things to consider here: the story being told, the telling of the story, and the reading of the story.
If your story is set before the name changed, then any dialogue or signs should show the old name. If it is set after the name has changed, then any signs should show the new name, but dialogue will vary depending on the person speaking - some may prefer the old name and keep using it, others may forget or not know it has changed, or just use it out of force of habit.
Depending on the narration of your story, you may choose between the two - for example, if the narration is in the past tense (someone is telling a story which has already happened, rather than telling it as it happens), then it may make sense to have the narrator explain the name change in an aside. Of course, you may then need to take into account when the narration is set. If the whole story is a flashback being told by a character in a narrative wrapper which is still before the name change, then you don't have that excuse or option.
Less commonly, you have the option of adding a footnote when the name first comes up, to explain (and, perhaps again later, if there is a large gap when the airport isn't mentioned) as an Author's or Editor's commentary. This is more appropriate for a non-fiction work, such as a history book
Dean called for a taxi. "Take me to Idlewild Airport," he said (This being before it was renamed to JFK Airport).
Dean called for a taxi to take him to JFK Airport - or, as it was called at the time, Idlewild Airport.
Dean called for a taxi to take him to Idlewild Airport1
1"Idlewild Airport" was renamed to "John F. Kennedy International Airport" on 24 December, 1964
I would usually say here that the main thing to maintain is consistency - however, as alluded to in the second paragraph, this may be consistency within characters, rather than the story. Which name the characters choose to use can give the reader a view of their mindset or personality.
Finally, consider your audience: you would handle this differently when aiming at teens or young adults, versus comfortably retired readers who remember the old name. Are they liable to get confused by you using the old name? Or, are they going to find it comforting and familiar, helping them to accept the setting and the era?