My understanding is that the name Gillis is a shortened version (presumably shortened during immigration to the USA) of the name McGillis, which was hails from Scotland.
I believe there is even a Scottish plaid and a coat of arms associated with the Gillis name. I have a paper copy of this information back in the house where I grew up -- if I get the chance next visit I will locate that, scan it and upload that information here.
Gillis is also the name of Jewish families originating in western Lithuania, especially in the town of Kretinga.
Controversy exists over the origin of the name Gillis. Some suggest that it derives from the patronymic Hillel, while another proposal bases the name on the more unusual matronymic Geula.
The association of Gillis with Hillel, meaning 'praise' in Hebrew, is given credence by Alexander Beider in his important study of Jewish names in Russia. The transferal in Russian of the letter H to G here shifts Hillel to Gillel and then to Gillis.
Alternatively, Gillis may well be a name given in honour of the matriarch of the family as a derivative of Geula, meaning deliverance in Hebrew. This name is often given to girls born during Pesach, alluding to the deliverance of the People of Israel from the hands of the Egyptians. There is usually a compelling reason to name a family after a woman's given name. Matronymics may be traced to a Jewish mother of heroic proportions, who probably overshadowed her husband, or my denote that a woman was widowed while young and then raised the fatherless children without remarrying. So the literal meaning of Gillis, according to this explanation, is the 'descendents of Geula'.
Gillis (ou Gilis) Provient du nom de baptême français GILLES sous la forme latinisée (sans doute à l'époque où les curés tenaient les registres de naissances en latin) nom fréquent en France en Ardèche ,dans le Sud-est ;on le trouve aussi en Lot-&-Garonne.Le nom Gilles vient du grec et signifie " protection " (Egide) - Saint Gilles ,athénien de naissance, installa un ermitage près de Nimes en France puis fit construire un monastère où il recevait les pèlerins qui allaient à St Jacques de Compostelle ,vers l'an 700.La cathédrale d'Edimbourg en Ecosse porte son nom .Il est vraisemblable que ce nom doit sa diffusion en Europe occidentale à un homonyme germanique "Gisil " = " le descendant de bonne race " que l'on trouve dans le nom Gilbert .