Before proceeding, I should note that I am fluent in Hebrew and a student of history.
Now, that I've note the above I'll proceed to address an error in the references to both names at this site.
The origin of the name Elizabeth (and respective permutations of the name in various languages) is the Hebrew name Elisheva ( אלישבע ). It could be loosely translated as "G-d's promise" (Eli=(my)G-d) and Sheva, from the word Shvua, meaning "oath" and interpreted also as "promise"
The origin of the name Izabel, Jesabel, Isabel (and respective permutations of the name in various languages) is the Phoenician name Izevel. Izevel married the king of the northern kingdom of Israel in the 9th C BC. That kingdom is now referred to as the "10 lost tribes". She came from a pagan background and was not respected nor well regarded. When written with Hebrew letters ( איזבל ), the pronunciation also means "Island of" or "remote place of" for the first part ( אי ), pronounced as "Ee" and "garbage" or "refuse" for the second part ( זבל ), pronounced as Zevel. The two parts together can mean "remote trash" or "foreign trash". This jives and conicides with the reputation of the Phoenician pagan woman who married the king of the Northern kingdom of Israel in the 9th C BC.
In no manner whatsoever, are the names Elizabeth and Izabel connected or derived from one another. The later permutations of the sounds of the respective words, occurring centuries later in Spain, etc. do not support the suggestion of congruence nor derivation. This should be noted as the two names are substantially different, of different origin, meaning and history.