The saying "Our family came over with The Mayflower" may well be something that you have heard from family members or from other people when they are referring to their family tree. But not every claim can be correct as only 10% of the US population are likely trues descendants.
Over time it appears that the saying has developed a general meaning rather than literal one. It is a way of saying that their family has been in the US since almost the beginning, regardless of which ship they arrived on or when. It can help people feel a sense of identity and patriotism, so let's not knock it too hard.
If you haven't looked into the legitimacy of a claimed connection before, it's not too late. There is a tremendous amount of information available through publications and societies to help you with research. A wonderful site for reliable information and links is http://mayflowerhistory.com/
It shows the full list of all 102 passengers and individual links to more information about each of them. If you think that 102 people is too much to research for a connection to your blood line, here's a spoiler alert: 45 of those passengers were dead by the first winter of 1620. Of the those that lived to 1621, it is documented that not all of them stayed in America or produced surviving children.
The Mayflower Society accepts members who can prove direct descent
If you have a list of relatives leading to a Mayflower Pilgrim and you want to learn if you are a descendant, check out their Mayflower Lineage Match page. They compare your list with the lines we have already approved, then let you know what you need to do to join the Society.
There are many reasons to join. They have Member Societies around the world in all U. S. states, the District of Columbia, Canada, Europe, and Australia. You can attend meetings close to home, and make friends with others that are also related to the Pilgrims. Learning about your heritage can be very emotional. It helps you to know who you are, how you ended up where you live, and perhaps what famous celebrities or U.S. Presidents may be your relatives. It also inspires you to broaden your horizons and travel all over the world in search of Pilgrim-related history. By joining the Mayflower Society, you are creating history for your family. The Mayflower Society will preserve your lineage records which becomes your legacy for future descendants.
If you have not got any family history research to enable you to submit to the society, then starting your family tree will be the first step. Start with your details and gradually go back over each generation looking for those ancestral connections.
If you'd like the assistance of a professional genealogist that provides a regular service to research and cite sources then AnotherLeafGenealogy.com may be a good fit for you.
If on the other hand you enjoy the hunt and have the time then sites like Ancestry.com and Familysearch.org may be the best starting place to finding that Mayflower connection!