I got the same laptop, and im also trying to figure out if it is possible to change the cpu.
I have already put more RAM in it, so it now got 8gb.
I have also changed the HDD to a 512gb SSD.
And I have installed "Linux Mint" alongside Windows 10, so I can choose at boot which one I want to start. Linux Mint runs much smoother and the cooling fan doesn't spin all the time like in Win10.
Oh and another thing that might improve the performance of the laptop is by removing dust from the heat-sink and changing the old thermal paste, (as it is quit possible that it might have dried out) and as a result the CPU overheats and the computer throttle down the CPU to protect the computer. (I haven't changed the thermal paste myself yet.)
Some CPU's are soldered to the motherboard and are therefore not changeable. But according to a SONY service/repair manual, the CPU in the VPCEH-series are not soldered to the board.
https/www.manualslib.com/manual/686993/Sony-Vaio-Vpceh-Series.html?page=13#manual
The VPCEH1J1E comes with a Intel Pentium B940 CPU, and it fits in a G2 (PGA988B) socket.
Here is a list of other CPU's that will fit in that socket:
http://www.cpu-upgrade.com/CPUs/Intel/Pentium_Dual-Core_Mobile/B940.html#sspec
But as far as I can understand, just because a CPU can fit in the same socket that dosn't mean the computer will boot with it, fx. the bios also need to be compatible with the CPU....
But according to this benchmark site it looks like some have run the benchmark on the VPCEH1J1E fitted with a "Core i5-2520M" and the "Core i5-2450M".
https://www.userbenchmark.com/System/Sony-VPCEH1J1E/10881
And btw you will need thermal paste to apply between a CPU and the heat sink.
Well that is what I have figured out until now....