Anyone who has started genealogical research has hoped to come across a famous ancestor. Maybe Daniel Boone was a third great granduncle or Martha Washington was a distance cousin. It is fun and a bit intriguing to think of the possibilities.
Then there are the family legends and tales handed down over the generations of being related to a well-known military general or to a great inventor. Such family stories do need to be researched and verified to see if there was any solid evidence and truth to the long told stories. If it is found to be true due to intense research, all the sources can be collected for future generations. If the story was stretched a bit, then the family has the actual family lineage with nothing out of place.
One method to start checking on a certain famous person and if they are related is to research that individual. It is very likely their family history has already been researched by someone. If not, their official biography may have enough information of parents and family hometown to get you started in the right direction. A good online site for biographies has some 25,000 biographies of the famous and infamous people. Another with some 30,000 biographies is Infoplease.
Locating any newspapers articles during the time the famous person was alive and well-known can provide additional family information. A newspaper obituary or write-up at the time of their death could be quite lengthy and include names of family and where they lived. With those names and places in hand you can better tell if there are any common denominators showing a relationship.
Check in the region, county or state the famous individual lived. The local genealogical and historical societies usually carry a large collection of any well-known people who came from their locality.
There is possibility in the ordinary research of your family line, you come across a name that you think might have been famous in their time. Take the time to investigate further by checking biography sites or newspapers. You may have never heard of a certain individual or that person might not be well-known in your time. Yet, in their lifetime that individual was very well-known and considered a celebrity in their own right. Many of the once most famous authors, inventors, generals, actors, politicians, industrialists, etc., have been lost over the decades of time.