How to get an opportunity to meet an angel investor or business mentor for free?

How? Is there a smart or efficient way?

SCORE is great as noted. You might want to check in with your local SBA office (www.sba.gov) as they may be able to connect you with a local mentor.

Angel Investors. Identify angel investor groups in your area (you can check out my posts on this subject if you're interested http://unchained-entrepreneur.com/new-rules-of-angel-investing/ ). Then look at their websites to see if they are hosting or participating in any events.

Also, find the individual membership of these investor groups and try and connect with them via LinkedIn.

Best of luck!

You can ask one directly over here.

try attending a business plan competition as a guest... The local one has an open semifinal fair and final pitch/presentation which is a great place to meet established business people and entrepreneurs

Look up SCORE if you're in the US, or try your local Chamber of Commerce for this kind of information.