You can say something as simple as "Thanks for connecting!" or even "It's great to meet you!" If you enjoyed an article they wrote or shared, you can say something like, "Loved your article!" or "Thanks for sharing your insight!" If you met your new LinkedIn connection in person at a 

networking event, mention something specific like, "It was great to meet you yesterday at the ABC Event!" or "Wanted to continue our conversation from yesterday."

Send your new connections a brief message to introduce yourself, or reintroduce yourself if you've previously met.  This will give them a refresher and will help jog their memory from when you last spoke.

In order to have a strong network, you need to constantly offer value to it. When you meet a new connection, make it clear that you'll be a valuable contributor to his or her network. You can say something like, "Please let me know how I can help you achieve your goals," or "I'd be happy to support you in any way I can.

It's important to build a relationship with this person before you ask for any favors.  They'll be more likely to help you out once they know who you are and that you'll return the favor.

Take a look at your new LinkedIn connection's profile and see what you have in common. Do you share a similar skill set? Do you volunteer for the same organization? Find something you share and start a conversation around it. It will make it easier for you to break into a real dialogue. 

Plus, it will help build a stronger personal connection between the two of you.