If the computer already has the firmware update from 2012, it would not accept it again.
And if you do not want to upgrade to 16GB, you could buy two 4-GB chips & run 8GB total.
There is (almost) no reason to suspect the unit could not accept 8GB, if not a full 16GB.
The best results are with matching memory chips of exact specification, from reputable
vendors who have tested the RAM they sell in the intended Macintosh computer models.
MacBook Mid 2010 - View Memory Options - OWC macsales
• MacBook: How to remove or install memory - Apple Support
I'd call the OWC macsales company (or email, if not in reach of their 800#) and ask if they
could help you to choose the highest amount of memory upgrade; and have the MacBook
serial number and sales order model number handy if they should require an identifier...
You can get the information without any commitment to buy anything from them.
If your MacBook is a MacBook7.1, it should have no problem using up to 16GB RAM, as
indicated by the OWC site, where they say the MacBook6.1 can use 8GB upgrade, and
the MacBook7.1 can use 16GB RAM upgrade. Note processor & other specs to identify:
MacBook "Unibody"
OWC tested to support Maximum Memory of 16.0GB:
MacBook "Unibody" (May 2010) 13.3" 2.4GHz (Model Identifier MacBook7,1)
MacBook "Unibody"
OWC tested to support Maximum Memory of 8.0GB:
• MacBook "Unibody" (October 2009) 13.3" 2.26GHz (Model Identifier MacBook6,1)
On occasion, they may list items at discount and also may have tested good & guaranteed
RAM that have been removed during upgrades, for less. Those deals are very limited.
You should be able to tell what the boot firmware version already in the MacBook7.1
is now, within the System Profiler or system information section, from the OS X.
The MacBook (Mid 2010) EFI Firmware Update 2.1 information & download is here:
• MacBook (Mid 2010) EFI Firmware Update 2.1
There is a chance the firmware was already updated in the past three years, evidence
of the status should be found with info from the Boot ROM version, system information
in the About This Mac under Apple menu, in Finder. Choose system report or profiler,
depending on the OS X version this access button may have a different name.
Check this Everymac.com page about the Macbook7.1 build Mid-2010 to see if it's yours:
http://www.everymac.com/systems/apple/macbook/specs/macbook-core-2-duo-2.4-white -13-polycarbonate-unibody-mid-2010-specs.html
In any event...
Good luck & happy computing! 🙂