The short (very short) History of Spike/Angel's fascination with name changes.

Angel:

Our sweet boy was born Liam, which in Ireland is the same as our English William. The name means "protection".

After Liam was turned he took the name Angelus, the Latinate for Angel, after his sister Kathy mistook him for the spirit of her recently deceased brother.

While a vampire Angelus earned the name "Scourge of Europe" after terrorizing town after town in the company of Darla, Spike, and Drusilla.

Angel dropped the "us" from his name sometime after coming to America in 1905. It has never been stated in canon exactly when this change happened.

Spike:

Spike was born William, which in England is the same as the Irish Liam. The name means "protector".

William was known, unfavorably, as William the Bloody in his lifetime because of his bloody-awful poetry. He kept the name after being turned, by torturing his victims with railroad spikes.

Shortly after being turned William took the name Spike. The most common theory that has come up in canon as to why is that a party goer before William was turned said he'd rather have a railroad spike through his head than listen to William's poetry, one presumes William obliged after being turned. He has been referred to as William by Buffy, Drusilla, and Angel after taking the name Spike.

Darla:

Although not her name at birth, Darla, the name given her by the master, means "dear one" or "loved one" and is of Anglo-Saxon derivation.

Darla said on more than one occasion she could not remember what her name had been when she was human and Angel confirmed that she never told it to him. It can be assumed that as a vampire Darla wanted to forget everything about her human life, and thus made no effort to remember the details of that life.

Drusilla:

Drusilla has roots in both the Roman language, as a derivation of the family name Drusus meaning "fruitful". It was also the name of Caligula's sister. The name may also have early Celtic origins, meaning "strong". Lastly, and most likely the origin of her human name Drusilla has biblical origins, and means "watered by the dew"?.

It is unlikely that Drusilla was named after she became a vampire, and much more likely that Angelus would continue to use her human name as a form of torture, a daily reminder of what she had been and the family she lost at his hands.