I have not had that problem with any of the players I have but, then again, I tend to rent only BR discs. If you have a DVD player that works for this, use it as I do not think there is a statistically reliable recommendation possible.
Buy regular Lemon Pledge in the yellow can and spray the offending disc, wipe it off with a lens cleaning cloth or super soft towel and see if that helps. Pledge fills in fine scratches and usually works like a charm for a few viewings. I use it on my eyeglasses, too for the same reason: fine scratches disappear.
Rental discs are notoriously scratched up whereas your personal discs are likely pristine and that's why they don't hiccup like the rentals.
Get rid of it!
I have a 2009 Samsung BDP3600 and I have the "exact" same issue. About 1 out of 5 new blu-ray dvds will not play. It drives me crazy. Furthermore the function keys, like skip or menu, have never worked. I have a cheap Sony blu-ray player as well and typically a dvd (either blu-ray or standard) that doesn't play on the Samsung plays just fine on the Sony. I had always thought it was the blu-ray DVDs that were the issue, that's not the case.
If the new Oppo BDP-93 was able to stream Pandora (last I looked this was not possible) the Samsung would be gone. The only reason why this is essential is that my wife is unable to figure out how to listen to Pandora via my Mac connected to the preamp; furthermore she would be equally challenged with a Roku box connected on a secondary HDMI port on the LCD. I know, I know....