J’ai du mal à la tête is technically grammatical but not idiomatic at all, unlike say j’ai du diabète, j’ai de l’eczéma or j’ai de la fièvre.

What is used when you want a name is:

J’ai un de ces mal de tête¹.

J’ai des maux de tête.

j’ai mal is equivalent to je souffre so mal might be understood as an adverb here (an invariable modifier word), although it’s more avoir mal which is a set expression. Similar ones: j’ai peur, j’ai froid, j’ai faim, j’ai chaud,…

We use it in J’ai mal à la tête, j’ai mal au cœur, j’ai mal au dos, and so on.

There is also an unrelated expression J’ai du mal à… that means It is difficult for me to do sth.

E.g.:

J’ai du mal à répondre à cette question.

¹ _{Although the Académie française doesn’t like it…}

I This word is here the noun "mal".

(TLFi) I. − Tout ce qui fait souffrir, physiquement ou moralement.
A. − Souffrance qui affecte le corps.
1. [De manière temporaire] Douleur passagère. Mal, maux de gorge, de tête; maux d’estomac, de reins; avoir mal aux pieds, à la poitrine, au ventre.
♦ «Eh bien! c’est donc le mal de dents, m’a-t-il dit.» En effet, il avait une violente fluxion; sa joue droite était enflée et fort rouge (Las Cases,Mémor. Ste-Hélène, t. 2, 1823, p. 223).

II No, it is not correct.

Addition suggested by user Silph‘s commentary: ‘1) if "mal" is a noun here, why is there no determiner? 2) why is it not correct? what is incorrect about it? is it ungrammatical? is it unidiomatic? etc‘.

1a/ This is without doubt the noun, and this is so because the adverb "mal" has nothing to do with pain. 1b/ The use of the article in French is not as nicely regular as one might think; in all sorts of expressions the article has been lost or is optional. In the case of that very expression, not so long ago Stendhal still wrote "Jai un mal à la tête."

(ref.) tellement l’air de lui sortir de la tête , et lui – même avait une apparence tellement 459 ( 41 ) se trouve incommodée ? J’ai un mal à la tête …

In fact, in modern French, it is enough to characterize the headache by means of an adjective or adjectival expression to make the article compulsory.

• J’ai un mal à la tête atroce.
• J’ai un mal à la tête si mauvais que je m’en taperais la tête contre les murs.

You could say that if there is no article it is because of an erosion more or less rapid of the form that has left no place for the normal form.
2/ Personally I do not find the use of the article so abnormal, especially as there are various intensities of pain depending on the cause of the headache. However, most people will feel that the article should not be used, and so, the usage without the article has given to the expression the turn of a somewhat fixed expression, or idiom, if you prefer; the use of the article, which seems normal has become unidiomatic.