"I have a one son" What's the rule explaining that mistake?
The simplest explanation would be that a/an = one, and thus there is redundancy.
The determiner theory would not explain "I have the one son.". Here "one" is an adjective = single or as OED says
II. Emphatic uses. 3.a. Designating exactly one, as opposed to two or more; a single ——. Frequently in negative contexts. Also preceded by a determiner such as any, no, some, the, this.
2002 Observer 28 July i. 24/3 A fine pub game in which you have to find the one London tube station that doesn't contain any letters from the word mackerel.
The word that describes the function of "one" or "a" in this context is determiner (Wikipedia). Determiners are different from adjective, and always come before adjectives.
From the above:
Determiners may be subcategorized as predeterminers, central determiners and postdeterminers, based on the order in which they can occur. For example, "all my many very young children" uses one of each. "My all many very young children" is not grammatically correct because a central determiner cannot precede a predeterminer.
Encyclopedia.com states that articles, ("a" or "the"), along with possessives ("my", "his") and demonstratives ("this", "that") are central determiners. Quantifiers such as "one", "two", and "many" are postdeterminers. So "I have a one son" does follow this rule.
I can't find the specific rule that "I have a one son" breaks, but if you look into the topic there are many such rules, some of them very specific, so we can safely assume this is its own rule:
The indefinite article "a" cannot be followed by a cardinal number (such as "one") as a postdeterminer
Note that this doesn't include "one" as a noun such as "I have a one of hearts" (referring to playing cards), or part of an adjective phrase, "I have a one ton truck".
Note that this rule doesn't seem to apply to any other central determiner. "I have my one son. I have the one son. I have some two sons."
Let's look at the possible uses of a/an/some.
They are used when you talk about a countable* thing, and you're specifying how many of them you are talking about.
- I have a son = I have one son (I may also have more sons, besides this one, but I am talking about this one for now)
- There are some problems with your approach = There is more than one problem with your approach
- An apple fell on my head = One apple fell on my head (perhaps another one fell later, but I'm talking about this one for now)
The reason I have a one son is invalid is because one son is not countable. It doesn't make sense for there to be two one sons, or even one one son.
(In response to comments on the question - I have a one son family is valid, because one son family is countable; "there are ten one son families living in this block of flats".
Another interesting case is I rolled a one [when rolling dice]. This is also valid - if you had 5 dice, you could say: I rolled five ones.)
* To be precise, 'a/an' are used before countable things, while 'some' can also be used before quantifiable things, like water; with 'some' you can state either 'how much' or 'how many'.