Answer
(Fatwa: 1301/1284/N=1433)
(1) I don’t know with surety to which language this name Eliza belongs and what does it mean? And no Shariah ruling can be written about the word whose meaning is not known.
(2) Its meaning is not good hence this name should not be kept.
(3) Adnan: it is the name of the great grandfather of the holy Prophet Muhammad (صلى الله عليه وسلم) at 21st grade of lineage, while in the history of Islam some scholars have the same name as well. And Ashhar means: the most famous one, well known.
Allah (Subhana Wa Ta’ala) knows Best
Darul Ifta,
Darul Uloom Deoband