The laptop comes with an i3 2350M cpu, socket rPPGA988 and HM65 Chipset.
My top choices are i5 2430M and i7 2640M. All have same TDP and support the chipset and socket mentioned.
The i5 should be officially compatible because the Toshiba Satellite L775 it comes with uses the same BIOS file as mine.
I have no idea if the i7 will work with BIOS but it fits the hardware requirements.
What to do?
I upgraded my L775-S7355 from the i5-2430M to an i7-2820QM and upgraded the RAM to 16GB and it has been working without an issue. Only recommendation I have is to get some 2000 grit sandpaper and wet sand your heatsink contact point to a light mirror shine to handle the increased thermal load along with some quality thermal grease.
Use this website and select Socket G2 and you'll find a list of every available and compatible processor with that chipset (I have already brought you to the HM65 compatibility chipset page, so you'll just need to select the socket) and I do believe the i7 3rd gen on the bottom is a mistake, so please ignore: CPU-Upgrade.com
Ram quantity and speed of what your unit will handle is based on the processor specifications, not the motherboard, so look that up before you order.
I upgraded a Satellite P745 from an I3 to an I7. Luckily, there was a youtube video that showed how to take that laptop entirely apart. Make sure that you have a compatible processor for the upgrade. I am sure someone here can tell you which one to use. Also, make sure you have a BIOS upgrade as well.
In addition, replace the hard drive with a fast one. My original Satellite came with a very slow rated one. People here will recommend that you get a solid state hard drive which is expensive. I didn't follow that advice because it isn't worth it on an old machine to upgrade to such an extent and spend that kind of money. You can get a fast regular hard drive that will work great. I upgraded my drive and cpu for grand total of $110 and the difference is amazing, You can buy a used CPU because those will work just fine. You can search for my thread to see what I did.
Go for it.
Only way to know is to try. I doubt anyone on here has tried a similar conversion.
But if the laptop came with the i5 as an option, then you'll probably be able to upgrade to it.
But if you can find the i7 for cheap enough you could try it, but I would say the chances of it working are less than the i5.