The word which is used in relative clauses:
- That's the car which I bought yesterday.
Notice that the relative clause which I bought yesterday is giving us more information about the car in question. So we have a noun car, which is being postmodified by a relative clause. We say that the noun car is the ANTECEDENT for the relative clause.
The relative pronoun what is very different from which. We only use the pronoun what when there is no antecedent. Look at the following example:
- That is what I bought yesterday.
Here there is no noun before the word what. So the word what means "the thing which". When we use it we don't name the thing involved. The problem with your student's sentence is that it is using an antecedent, the word one with the relative word what. In the type of standard English that you are promoting, the word what cannot occur with an antecedent.
Take any given sentence which this might occur in:
- I like the ball what is blue.
If you ask yourself:
... the answer should be:
If they use the right relative pronoun, there is no antecedent 'X' to be able to ask which X. This means the sentence was grammatical in the kind of standard English you're aiming at. Consider the sentence:
You cannot ask in response to this:
This shows the sentence is fine. You cannot correct it with which and so what is correct.
Having said all of this this, this is definitely only something that you would want to correct in writing. There is no point whatsoever trying to correct native speaker's speech. All the evidence shows that this is a complete waste of time and will not work. In addition, you run the risk of damaging you student's self-esteem and making them feel bad, and all for something that is not "fixable".
Although you may not have heard it before in your local area, this definitely is a varietal feature.