The name Elisabeth (and its variations) is biblical (Elisabeth is the mother of John the Baptist). The Greek name Elisabeth derives from Hebrew. The particle "Eli" means "God", and altogether name means something like "God is plenty". This biblical context explains why the name is known in many linguistic variations across the world.
I respect this spelling of Elizabeth as a variant translation and as the French and German form. It's very pretty, and a little more unusual. I probably would just go with the normal form, Elizabeth, though.