This is a bad RAM slot. One RAM slot is BGA, the other is through-hole. Read up on BGA soldering, and then through-hole soldering. Look at some pictures of each so you can tell immediately upon seeing something whether a component is soldered to the board using BGA or through-hole soldering..Then tell me which RAM slot is BGA, and which is through-hole.
Put one RAM stick in the through-hole soldered RAM slot, and no RAM in the BGA soldered RAM slot. The BGA soldered RAM slot often &&^&* up and results in what you are talking about here. There is no economical repair for this.
The place that diagnosed your logic board as the bad part was correct. The place that charged you for new RAM and hard drive are retards that have no place repairing consumer electronics.
The proper fix for this is to max out the working through-hole RAM slot and call it a day. crucial.com has a really easy application that you download that tells you exactly what RAM you need for your specific model computer, a good warranty, and excellent prices. Just max out the working slot and call it a day, and try to get your money back from the idiots you paid.
8 GB of RAM is "enough" at this point in time, and is now the limit of the board with only one slot. Do not replace the board, just so you can go to 16 GB of RAM, this is a total waste of money. Just stick with 8 GB of RAM and use the saved money to get an SSD. Or even better, two SSDs in RAID 0 - your machine will scream and cease to feel "broken" with spinning color wheels everytime you try and load up something intensive.