If you're thinking, "um, Isabella isn't in the Bible.....is it?" you're sort of right.
Let's start with Elisheva. It's the name of a couple characters in the Bible, most notably the mother of John the Baptist. You probably know her as Elizabeth. When the Bible was translated into Greek, Elisheva became Elisabet, and then Elisabeth in Latin and English (in fact, many of the names we think of as "Hebrew" are actually Hellenized and/or Latinized forms).
Provençal had problems with that final -th, however, so in Southern France, the name became Elisabel.
It then spread to Spanish and Catalan areas, and then went through something called hypercorrection. Hypercorrection is when linguistic rules are overapplied, particularly to a word or name from a different language. In this case, Spanish speakers may have interpreted Elisabel as "el Isabel" ("the Isabel"), and dropped the non-sensical masculine 'el'. Alternatively, Catalan speakers may have heard Elisabel as "e l'Isabel"("and the Isabel"). Or both could have happened! But for whatever reason, Elisabel became Isabel as it left Provence.
Isabel caught on like wildfire in the Middle Ages, becoming common throughout Europe. It was then re-Latinized (it wasn't exactly recognizable as a form of Elizabeth if you didn't know Provençal!) to Isabella.