what prompts bilbo to name his sword? Why is this a significant moment in the narrative?

Bilbo was proud of being able to kill the first spider without help from anyone,

"Somehow the killing of the giant spider, all alone by himself in the dark without the help of the wizard or the dwarves or of anyone else, made a great difference to Mr. Baggins. He felt a different person, and much fiercer and bolder in spite of an empty stomach, as he wiped his sword on the grass and put it back into its sheath."

"I will give you a name," he said to it, "and I shall call you Sting."