What is Pip's Christian name? How does he come to be known as "Pip"?

To answer this question, take a look at the opening of chapter one. According to the text, Pip's Christian name is Philip. Although Pip never knew his parents because they died when he was an infant, he knows that his surname is "Pirrip" because he has seen it written on his father's tombstone. This suggests that Pip's guardians, Mr. and Mrs. Gargery, do not talk about Pip's parents nor call him by his proper name.
The transformation from Philip Pirrip to Pip took place when Pip was a young child. He struggled to correctly pronounce these names together, as we learn from the text:

My infant tongue could make of both names nothing longer or more explicit than Pip.

As a result, Pip simply called himself "Pip," a name which is much easier to pronounce than Philip Pirrip. Over time, other people began to use this name and so it stuck.