Answer
A name of a person has great meaning it affects ones soul and ones body and it is the channel which via it flows one’s life (
chayus) rations and provisions, it has the has the ability to influence his present and future therefore it is very important when giving a name to a child to choose a proper name which will give the child proper life – chayus.
[1]
The Arizal (kabbalah) teach us that when the parents choose the name for their children g-d places the name which is needed for this child in their mouth , indeed they are considered prophets as giving the name is truly a small prophecy.
[2]
Therefore, giving a name is to be a mutual agreement of both parents together (and if they cannot reach a joint decision they should consult with a local Rabbi.
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With this in mind, parents will name their child after a tzaddik righteous man as t
[4]o draw down special spiritual forces – light to the child. And on the other hand, parents try to avoid giving the name of a wicked person to their child as not to draw forth to the child negative forces.
[5] So to great Chassidic masters emphasized not to give gentile names as the child can g-d forbid receive life from the forces of klipah.
[6]
Once a name was already chosen and given, we do not remove as the life comes via that name as kabalisticly that can be a danger. yet in very certain instances where there is a great reason we can actually add a name to someone (although this only done with proper rabbinical consultation as one might mistakenly make it worst) for example if one is ill in order to change or add to their mazal a name will be added for example by a male the name chaim or rephael (life , or g-d heal) and likewise by women the name Chaya (life / living) can be added.
When one does not have a Jewish name then it is very important that they receive a jewish name as the proper divine life comes via ones Hebrew name as the true life of everything is via its Hebrew name
[7] we usually we look for a name which is similar to one s preexisting gentile name for example one with the name Mary will take Miriam, Rebecca Rivka, Leon Aryeh or Leib.
In your particular case where the name is actually Isabel which has a double negative as not only is it a non- Jewish name , additionally it is the name of the wicked queen of Spain who exiled the Jews in 1492.
With this in mind it is definitely proper to take a Jewish name Though this is totally depending on your little prophecy.
It is important to add that our rabbis teach us that the Jews where redeemed from the exile of Egypt in merit of the fact that they did not change their jewish names to the Egyptian names which that truly teaches us how important that is
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