An unusual name to research is a genealogist's blessing. What are the chances that more than one person by the name Audio Science Clayton, Fifi Trixibell Geldof, or Moon Unit Zappa has ever existed? Some parents give their creative genes free reign in choosing their child's name.
For more common names you need to add additional facts, such as birth date and place, parents names, religion and more to establish the person's identity uniquely. It's the establishment of the unique identity, not the rote accumulation of additional facts, that should be the driving force in deciding when you have enough information to proceed further along your research path.
Naturally, some of those facts will be found without much effort from your base source and provide clues in pushing research further back in time. Just use your judgement before going to expensive lengths to accumulate evidence which will in all likelihood be redundant when your time and money can and must be better spent.