The first form is the most usual one.

Here sens de l’humour is seen as something you have or you haven’t. It is assumed there is just one sens de l’humour.

If you say il a un sens de l’humour, you mean the person has something that looks like the sens de l’humour but isn’t the mainstream one. That would be more like:

Il a un sens de l’humour particulier.

As a complement to user jlliagre’s answer I’ll add the following.
It is possible to speak of "un sens de l’humour" on the condition that "sens" be modified by an adjective; however, few adjectives might be idiomatic. The adjective "grand" is.

• Ils ont un grand sens de l’humour. (ngram, ref.)

As illogical as it may seem "un petit sens de l’humour" is not said; it seems that if some people are deemed able of possessing that quality to a great extent, then there should be others endowed with the possession of it in moderate quantities, but no, apparently this is not so: either you possess a normal understanding of humour or you possess it to a great extent or not at all; here is then another mystery of French linguistics to which one should not yet despair of finding a satisfying explanation despite the apparently irremediable nonsensical situation.

It’ll be possible to speak of a very mediocre sense of humour using this process of adjectival modification, the idiomatic adjective being "piètre".

• (ref.) Le silence se prolongeait, devenait insupportable. — Je plaisantais, Will, répéta-t-elle. Mon piètre sens de l’humour m’a encore joué des tours ! Désolée.

I suppose, from a certain experience that I have of the French language, that in order to supply a replacement for "petit", which as well as not being in Google books does not seem idiomatic to me, one can use "modeste", but I have no supporting reference.

• Un modeste sens de l’humour ne lui fait certainement pas défaut mais elle avait tendance à vouloir trop en faire.

A careful search will probably yield other useful idiomatic combinations.