You often have the opportunity to discover connecting points prior to a planned or chance encounter with a person you want to meet. People typically research a company ahead of a sales call or a job interview. Plan ahead with your people research too.
Discovering a connecting point to an individual(s) you meet, will often make for a more relaxed, possibly even friendlier mood. Perhaps you both came from the same hometown or state; attended the same university; know some of the same people; share an interest. Maybe you find something particularly special about the person.
Be careful though not to cross that thin line between connecting and stalking. Choose wisely the biggest impact connecting point you share from all the information you learned. DON'T say everything you found during your conversation. You could frighten the person and transform the mood twice; first from tense to friendly; then by saying too many much, to suspicious and a lost opportunity. Save the extra connecting points for future interactions with that person.
If I want the option of using several connecting points, I will mention the source of my discovered connecting point, like a newspaper story. I could then reference anything in that article without sounding like a stalker. Be careful again here that you don't say personal connecting points from additional sources. The beware-of-stalker alarm will start blaring.