Nevertheless, it is true that significant numbers of Yemenis participated in the Arab migrations to North Africa during the Islamic era, and it's not inherently implausible that some should have brought their languages with them. In fact, I just came across what looks very much like a South Arabian loan into the northwestern Libyan Berber variety of Zuwara (At Willul).
In Zuwara, the usual word for "father" is baba, as in many other Berber varieties, but in a few collocations such as əg tíddart n ḥíbi-s "in her father's house", a different term ḥibi is substituted (Mitchell 2009:303, 341). This word is unlikely to be proto-Berber, since proto-Berber did not have a phoneme /ḥ/ and since it is quite unusual within Berber. And as far as I know, it is not used anywhere in Arabic (although Libyan dialects are not that well documented). One could try to link it to ḥabīb-ī "my beloved", but that would be phonetically irregular and semantically unlikely, since this term is normally used in the context of romantic love or of a child by their parents.
However, the normal word for "father" in Mehri is ḥīb "father" - ḥayb-ī "my father", ḥīb-as "his father" (Watson 2012:149). In fact, Mehri adds this ḥ prefix to a number of kinship terms: ḥāmē "mother", ḥabrē "son", ḥabrīt "daughter" (ibid), as well as a number of other common nouns. Its function is to mark definiteness (ibid:64). But no such definite article has ever existed in Arabic or in Berber, so the only possible explanations for the similarity of Zuwara ḥibi are pure coincidence or borrowing from Mehri into Berber (perhaps via an Arabic dialect?). It will be interesting to see if other cases turn up.
And as long as I'm talking about Libyan Berber, I really ought to mention Marijn van Putten's new book A Grammar of Awjila Berber (see his announcement at Oriental Berber).. This careful analysis of all the unfortunately limited data available on the very unusual Berber variety of Awjila, in the far east of Libya, is an important resource for Berber historical linguistics. I hope that things settle down in Libya soon enough to make a fuller description possible, but for the moment, this work appears unlikely to be superseded.