The name “Nathaniel” is one of the few things given to him by his parents. I hear the name Nathaniel means something like "God's grace" or "gift from God" in Hebrew. (Natan=give, El=god)(It is ironic that his parents, who gave him this name, abandoned him.)
It is something that Nathaniel himself quietly cherished, and it also a name that stands as proof of Mrs Underwood’s love for him. Had she treated Nathaniel as less than human, like the other adults did, he might have forgotten his own name at an early age.
But even while Nathaniel was treated as a nameless child by his master, Mr Underwood, she went on loving him as if he were her own son, continuing to call him by his forbidden name.
As a result, she made the grave mistake of letting Bartimaeus learn Nathaniel’s true name, yet it goes without saying that this very name is what maintained the unique bond between Bartimaeus and Nathaniel.
Nathaniel managed not to become entirely “Mandrake” because Bartimaeus protected Nathaniel’s true name, and, unintentionally, that name was passed on to Bartimaeus through the love of Mrs Underwood.
Moreover, the name “Nathaniel” now holds the potential to remain for thousands of years through Bartimaeus (this is Stroud’s official statement). I believe that for Bart, this name Nathaniel could literally be a gift from God. At least, there is no doubt that this name has become one of his amulets for the rest of his life.
I deeply love this peculiar form of affection that has been inherited in such an unusual way.
(And by the way, when I learned that the name “John” means "mercy of God", I was so cynical that I shut my mouth.)(I know that people in that world don't follow a particular religion, but still.)
I didn't check it properly, so my English might be wrong! Thank you for reading.