Hello,
When is a science not a science? When it is biological!

GedMatch recommends over 7cM if you are using that criteria alone. The smaller you go the easier it is to get a match with anyone. I think I read elsewhere that 3cM gets that kind of result.
However, your percentage match may also matter. By trial and error initially, I realised that chasing links of 0.3% and less was a similar exercise. Yes, you can make connections but you are going way back and unless you have a really good, expansive tree to use, it is going to be a long job working out where it came from.
If you have an already well researched DNA and can place segments of DNA with known relatives then it is possible to go smaller (DNA painting comes in here)
One technique I use is to multiply the cM value by the number of SNPs, the bigger the score the more likely your going to find someone that touches your tree because both of you are likely to have known relatives to compare.
The thing for me is that the matches I want are the really small matches because I am back in the early 1800s and I have to find big matches to work their tree back to mine in some way.
Something I would recommend, as someone did already, is try to look for any match with given surnames off your tree. Obviously Smith might be a long shot.... However, also notice locations?
I have 12% Scandinavian genes as a surprise and one of the things I have been trying is collecting all the Scandinavian names, Johnson is the obvious name that is not in my tree. One thing that comes out of this is you will get many people from your 8000+ matches with the same surname - might be worth asking if a name recurs again and again. I have some notable links to people called Tucker with no idea where it comes from but it appears they have Scandinavian genes too!
When looking at possible matches I often use any combinations of these to make decisions about a letter of enquiry.
As the man said, "..it depends.." Make some fuzzy logic judgements on the items above and use your intuition too. It is isn't about science as such at this point?
One of the things lacking in DNA research is a how to guide on what to do with your results. Bits and pieces are given, but there is not an overall strategy as yet.
Hope this helps and please ask if I confused anyone?