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.