I setup a clean install of Linux Mint 19 Cinnamon on a 2007 Sony Vaio laptop with Core2Duo processor and 2 GB of memory. It has the original 160 GB HDD. I set the computer up for my parents and it works fine, but occasionally, it seems to lag a bit. The most demanding task that my parents will throw at it would be watching a Youtube Video at 1078 x 768 or whatever the resolution laptops of that era used. 99% of the time it will only be used for web surfing. Would installing a modern SSD remove the lag or is my bottleneck somewhere else? I'm thinking the processor should be sufficient and the minimum requirement is 1 GB of memory, 2 GB recommended. If it's just an SSD, I can order a small one, I just don't want to put more than about $50 into it since it's such an old laptop.
"Seems to lag a bit" is quite a wide definition. An SSD will speed up start-up of applications and responsiveness of the system when the HDD is under load. You can tell pretty easily if the HDD is slowing the computer down by listening to the seek noise. 2 GB RAM is also not much these days, so it also could happen that you'll hear the hard drive being busy because it's swapping to disk. Linux Mint MATE should be less memory hungry.
If the computer stutters when playing video, that would be a different problem, it would mean that the CPU is too slow to decode it and the GPU is not helping because it's either too old/weak (doesn't support decoding modern video formats) or it potentially could but Linux drivers don't support it.
You can see CPU and memory usage in a program called htop (can be installed from the repositories), that can also tell you where the problem could be. In any case, don't expect any miracles. I've been running a very highly clocked Core 2 Duo with 8 GB RAM and an SSD on a motherboard from 2007 and sometimes could not figure out why it was so slow compared to a nimble laptop with 2014 tech. It's just... old :)