"What is the best way to determine whether a stranger on LinkedIn would be an ideal contact during a lengthy jobs are?"
The language in this question just makes me want to go crazy?
I'm going to avoid "the lengthy jobs are" part of this question. To me that is irrelevant to answering the question. What is important is how do you determine whether someone is able to help you? You can't guarantee outcomes, but you can look at probabilities.
If you are targeting organizations, you want to target someone who's in authority at that organization. You are not looking for the CIO if you are a programmer. You are not looking for the president if you are an accountant. The power disparity is too great; they will kick you down through the chain of command very quickly.
What you want to be doing is building relationships. This is not about being transactional. That is what you criticize third-party recruiters for and then you try to go out there and become transactional yourself. What you need to be doing is looking at people who are in positions of authority within an organization that you are targeting so that in this way, when you do outreach to them ("hi! I am looking for advice."), Think of an informational interview.
"I'm looking for some advice. I am a such and such. You are further along in your career trail than I am. How did you get to where you are? Can I schedule time for a Skype or a phone call or an in person interview with you?"
Your preference should be in person or Skype to chat with them for just 10 or 15 minutes and what you are doing is trying to build a relationship. If you think you are going to get one connection request, what acceptance and think you're going to get hired, you are deluding yourself and you have been going for your search all wrong.
Again, this is the slow road to using your LinkedIn network. If you think about it, if you are 24 years old and you are thinking ahead to being 34 years old (I note that seems like a long time from now) or 44 or 54, the relationships that you build now can be very pivotal. I am older than that; I'm now 66 and worked in recruiting for more than 40 years.
From my vantage point there are a lot of people that I have known from the beginning, with whom I have develop relationships, who have hired from me, they have taken jobs from me and we have become friends over the course of time. This is really how you use LinkedIn. It is building your network of connections.
If you are in a long jobs are and have been looking for a long time without getting any results, you've got bigger problems than just simply the LinkedIn connection. There is something wrong with how you are conducting your job search. Your resume may stink; you may interview terribly. You need coaching along those lines, and I will simply say that I have a lot of videos that you can watch at JobSearchCoachingHQ,com that will help you with that
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