Honesty is (once again) the best policy

It seems a person can benefit the most from his/her Linkedin connections by simply telling them what he/she needs, and what he/she can offer.

For example, you might want to list the kinds of information you are looking for, and might want to list the kinds of information you currently have (bookmarks, white papers, articles, etc.) so that your connections can more easily "trade" with you.

You might also want to state your career goal, and gently/politely ask people to forward information to you that might help you get closer to that career goal.

Last but not least, you can provide examples of how you were able to help people in the past, so that your connections know what you are able and willing to do for them.