I'm successfully installed Windows 10 Pro 64 bit on my HP 6510B 4GB RAM and 240GB SSD with significant good performance. My next upgrade is to upgrade my T7300 cpu to T9500 CPU and hopefully if someone from HP can confirm if this is possible to upgrade the RAM to 8gb.
Kindly check if this is upgradeable, I found in this website they are selling specific 8G ram upgrade kit for HP 6510B model.
This is a peer to peer forum. No one from HP will confirm that 8 GB of memory will work in your notebook.
What I can tell you is I have two HP 6910p notebooks with the T7500 processor and the same chipset as your model and they both run 2 x 4 GB of DDR2-6400 memory just fine. The only thing is if you don't know it already is the mobile Intel 965 chipset only runs DDR2-6400 memory at 667 MHz, not 800 MHz.
I would not be able to confirm the memory you want to buy will work.
This is the memory I bought for mine many years ago...
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