Avery personal and unique Armenian gift would be to have someone's name handwritten using the Armenian alphabet, hand painted and hand inscribed on a ceramic tile from Jerusalem.
This traditional art goes back to Kutahia, Turkey, a city south east of Istanbul, where Armenians lived and worked as ceramic artists and potters since the 14th century.
In fact, there are ceramic wares using Armenian alphabet handwriting from this period in Kutahia. These ceramic wares with Armenian writing were usually presented as gifts to priests serving in the local churches in Kutahia, or lay people, in appreciation of their service to the community. One such ceramic ware with Armenian alphabet handwriting is displayed in the Victoria and Albert museum in London, more about it here.