The genetic study "Ancient MtDNA Sequences and Radiocarbon Dating of Human Bones from the Chalcolithic Caves of Wadi El-Makkukh" by M. Salamon, Shay Tzur, et al. in Mediterranean Archaeology and Archaeometry vol. 10, no. 2 (January 2010) on pages 1-14 included a carrier of mtDNA H6 from ancient Israel!
Labeled sample WM30, he was a male who died sometime between 4315-4045 B.C.E. (per Table 1) and was found in a cave at the Wadi el Makkukh site in the Judean desert, near Jericho.
There are 4 possibilities for what branch of H6 he belonged to, because he carried the extra mutation 16311C. The authors of the paper wrote on page 8 that he "most likely belong to H6a1a1 haplogroup". We cannot determine this for certain because the sample was only partially sequenced.