Your good name is basically your first name.
It's a throwback from our British colonial days... where a gentleman would ask another who is not of acquaintance and would like to be friendly - "May I ask your good name, sir?" or something on those terms.
And if they ask for your full name - well you tell your full name. In India, it's preferable to use "First Name" "Last Name" in less formal or official situations like signing the guest book at a hotel, or introducing yourself to someone officially.
You may include your middle name if you're filling out some sort of official form or application - only if there is an entry field asking you for it on the document.
EDIT - clarification - linguistic influences of the Queen's English / Oxbridge English on present day Indian English
As obvious English is not India's native language - we adopted it due to British colonial influence and their efforts in education. It's worth noting education, particularly schools and institutions which had an English curriculum were usually catered to the Indian elites (read: princes, children of rich businessmen/ zamindars, etc).
This strata of society interacted more with the British elite who were basically the ruling class at that time. As was the norm - most English people in this group were all titled (duke, earl .. at least a knighthood) or were highly respected. If you read the works of the English author's of that time - you would find many instances (can be a bit exaggerated but still hold true) of how people would interact in formal and informal social occasions.
Indian educated adopted these mannerism from them - especially the Bengali's - people from Bengal have always been academically inclined and many of them used to be office bearers in British companies and institutions. Contrary to the general portrayal of the colonial times in media, The English were generally polite and well mannered and treat such educated Indian officers with a degree of respect.
Obviously Indians educated in the English language would follow the mannerism taught to them; at least while talking to their English superiors. As time progressed - these mannerism flowed down the rungs of society. With the atrociously long names given in many Indian communities and the fact that most of our legal and governmental documentation system still bear a huge influence from the old English system - the Idea of "what is your good name" evolved to it's present day in India.
I would go on to explain the confusion of the "naming schemes" in India - but I think Fixed point elaborated on that quite well - Though the Bhalo naam concept isn't really isolated to Bengalis only.
tl;dr: - Indian naming system doesn't have a good name concept - we adopted this concept from the English language and naming system.