While it is beyond the skill level of most people, the RAM slots can be replaced. Unfortunately if you want to have both RAM slots available, you will need to have this repair done and you don't want to replace the motherboard. The average person is better off running a single large memory module, since the repairs that are required to fix the board are very difficult. The downside to this is you can run half of the memory you could before, but it's the easy fix to deal with a bad RAM slot.
Before you replace the RAM socket, check if the northbridge has an issue and use a completely different set of memory. If this works, both memory modules likely have a problem and replacing the memory was enough to clear up the problem. In most cases, the problem lies with the RAM instead of the socket or chipsets.
However, if it is a bad northbridge, then you will need to decide if it is worthwhile to have the chipset replaced. It most likely will not be and you'll be better off using the machine with one RAM slot for the rest of it's life or selling it for parts.
If you have tested both of these components and find notheing wrong, it's a memory socket problem. The difficulty of sourcing the part and doing the repair depends on which socket is bad but the top slot is much harder to replace then the bottom slot. You also won't be able to find sockets that use the Apple OEM color, since Apple doesn't like 3rd parties being able to repair their stuff. You will also lose the serial number sticker on the board if the top slot is bad if is is not transferred or it is burned, so try to keep it intact if possible. It's best to remove this and put it on wax paper to preserve it.
In terms of sourcing the part, this machine uses a uncommon socket style. You're going to have a harder time finding the correct part but it's not impossible. The only thing that's special about the bottom slot is the plastic is black. It's not important to match this, so it's okay to get a socket that uses the wrong color plastic.
Since the top slot is so sensitive and easy to damage, it is absolutely imperative a quality hot air station is used to avoid damage. Do not use a cheap station since the odds of damage are much higher. It is also a good idea to use a lot of flux to make the solder flow quickly and minimize the heat exposure to the sockets.
If you have not performed such a repair and want to try, spend some time seing which technique works the best and offers the least amount of risk to the top RAM slot on the board. If you determine the job is above your skill level, don't mess with the board and send it to someone who can do the job correctly.
If you're able to repair the board, here's a link to the guide to remove the motherboard: MacBook Pro 13" Unibody Mid 2010 Logic Board Replacement
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