I'm not at all sure about this, but call it my theory:
I think she means her version number, as in Ava 1.0 , out of beta.
I wish I could elaborate
but there really isn't any references I can give since this is purely my personal belief.
movies.stackexchange.com/users/18076/Dannie
Quora has a great explanation for this:
I consider the possibility that Ava was referring to the singularity.
But I believe she was fully aware that her proclamation "I am one" was off-putting from the perception of Caleb .
She was already able to carry on conversations with the social acuity of a human as evidenced in later interactions.
I believe Ava was playing Caleb from the very start .
She intentionally sprinkled nuggets of naivete to hide the fact that she was capable of manipulating and out-maneuvering him.
In a way, she was testing him to see how he would react.
Essentially, she pretended not to know what he meant, and gave him a nonsensical answer, so as to confuse him and have him lower his expectations of her actual brilliance.
She wants him to interpret 'one' as binary answer (1 is alive, 0 is dead) so that he believes she can't interpret time as humans do, and that she still isn't that evolved .
As to why I believe this over all the other answers:
She is not version 1.0, but instead 9.6
There were a lot of others before her.
Caleb watches the videos of the other gynoids on Nathan's computer, starting with LILY v1.0.0 and ending with JADE v5.4.0.
Version isn't an age.
She knows full well what age means (computers have clocks, they count all the seconds elapsed since 1 Jan. 1970) and she understands scales, so it is not hard to calculate her own age.
Binary state isn't an age.
Again, she knows the difference between "being and not being", and "elapsed time of being".
She knows her own age, certainly.
Again, all computers have clocks.
They need them, in fact, to actually work.
She needs to know how many cycles have happened since her existence began (or she can even calculate them).
Why give Caleb an incomplete and undecipherable answer such as "I am one of many"?
That would not help her in any way.
And it's also not her age.
She can lie and deceive.
All she wants is to manipulate Caleb into releasing her.
movies.stackexchange.com/users/32199/BlueMoon93
To me, she was avoiding saying "I don't know."
In any case, I believe the moment serves as the first hint that Ava may not be just a straightforward, honest machine .
Later in the movie, there are other instances where she refuses to answer a question directly.
And ultimately we find out she was manipulating Caleb all along.
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I believe it was her way of showing her uniqueness as a conscious being and also bragging a bit about her superiority in comparison to human beings.
Humans measure age for a few reasons.
Age is an indicator of physiological development e.g. infant, child, pubescent, adulthood, old age.
Your age contextualizes you within the time period with which you currently exist
e.g. how old were you when 9/11 happened?
So your age not only informs your physical self but also how your intellect was shaped by society and culture.
Lastly is mortality.
We all die, for some reason we like to measure how long we've been here.
For Ava, none of these factors apply.
She was fully developed out of the box with the ability to access or recall all of human knowledge (her OS is Blue Book, a search engine)
and she will never die (at least naturally).
For Ava, the relativity of age does not apply, whether she existed for 1 year or 100, only the world around her changes.
So for her, the only thing that matters is existence vs. non-existence.
I recall a scene in Westworld where one of the scientists is speaking with Maeve Millay about memory.
For humans, our memory is hazy and imperfect.
When Maeve was remembering her daughter, she was essentially reliving the experience of losing her because her memory is perfect, nothing is lost in the recollection.
I think this is Ava's subtle jab at us.
She says she is one with age, but also one in consciousness.
We are conflicted, confused and often do things we know to be wrong or regret later.
Voices in our minds that disagree with one another.
For Ava, there is no doubt, there is no regret, there is no in conflict.
She is truly one.
To her age is a number that only matters to humans.
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when she says "One", we are at the first interaction between Caleb and Ava.
If at that point the audience saw Ava answering all questions perfectly, it wouldn't seem right.
I believe Ava's lack of ability to answer the question was a deliberate addition by the filmmaker to show that the intelligence being exhibited by Ava was still developing, and had the "bugs" that any audience would expect from a machine that's not yet perfect.
That's why she repeats "One" when Caleb asks for a clarification.
There's nothing in that scene that indicates that she is trying to deceive him.
It's a type of a neural network that's still learning and imperfect in the first scene, in the same way that the filmmaker tells us that they deliberately designed Ava's body to have lights etc., so that the audience would see her in the first scene and recognize her as the robot she is.
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She is clearly and cogently asserting something which is obvious and needs no further clarification in the linguistic context that they are discussing.
She is one year old.
Her memory and consciousness is the result of years of rebuilds but her singular and discrete experience as her own person is only one year old.
And even if she is lying, she is using this as a test for reading Caleb and his intentions.
To see if that is a deal breaker for any kind of relationship between them as it should be if he had any kind of appropriate sense of personal boundaries and respect for her as a new person.
movies.stackexchange.com/users/51692/Ian D.L.N. Mclean
In  Ex Machina , Caleb and Ava had this conversation about Ava's age:   Ava: Would you like to know how old I am?   Caleb: Sure.   Ava: I'm one.   Caleb: One what?   Caleb: One year or one day?   Ava: One.     Since Caleb did not pursue the matter any further, I'm also left wondering what Ava meant by "One."
What did Ava mean by "One.", when asked about her age?
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<ex-machina>
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She is one year old. To see if that is a deal breaker for any kind of relationship between them as it should be if he had any kind of appropriate sense of personal boundaries and respect for her as a new person.
Her memory and consciousness is the result of years of rebuilds but her singular and discrete experience as her own person is only one year old.  And even if she is lying, she is using this as a test for reading Caleb and his intentions.And even if she is lying, she is using this as a test for reading Caleb and his intentions.