Great little media center you have. This is full spec of my secondary PC.
Intel® Pentium® Processor G2030.
IPMSB-H61.
4GB RAM.
500mb old WD Black.
240W cheap PSU.
My first build is on my signature. Seems like I would just use it for heavy task occasionally, that's why I wanna make the second PC more decent for everyday light uses, or media center like you have.
It can take an SSD you might be able to make things load faster, open more responsively, in general if its not had anything done to it for a while the drive is probably a lot slower than it would have been, and any normal HDD should make it feel more responsive opening windows and programs. You got any drive you can clear and spare from your main rig? do a fresh install of windows on a newer drive and give it a go, nothing really to lose, if a HDD makes it run better try an SSD, if the SSD doesnt make much of a difference than the HDD use the SSD in your main PC. it isnt something you will waste. i think you will notice the main difference in boot times, startup and shutdown will seem much quicker.
That should apply to both systems in all honesty, the thinkpad might appreciate it more than the desktop, being bumped about over the years wouldnt have done the platters many favours. I think its worth a try but i wouldnt be holding up massive hopes, it will make it feel alot newer for sure but generally wont make programs run any better or worse, once things load from the HDD to system ram its going on that, an SSD is more reliable in a laptop it can take a little more stress so maybe worth it just for that reason alone.
4GB of ram in todays operating systems might not hold up too well, it might iv not had that little for a while myselself, might not be a great idea to install windows 10, tho its generally more stable, could be a bit of a toss up what will work best, if you have time and patience try various installs going back to XP. or have a look about for some DDR3 going cheap. im not sure if you can upgrade much on the thinkpad but that would be soDIMM and your normal ram is just DIMM DDR3 8GB will give you a much better experience
while we talk cheap that old PSU might not be as reliable as it used to not be lol i wouldnt put much trust in it when you can get something with a name you can half trust, a cheap bequiet for $30 is a nice bit of peace and mind. you dont need any bells or whistles on it but a warranty is always nice
That'd be how i'd be looking at things if i was you anyway, i hate saying Do this Do that, id rather day how i'd go about things and why,
My big issue with things is once you start needing to replace too much you might as well take a step back and look at other routes, an SSD and PSU Ram all in will be what $100 (im in the EU here) a raspberry pi might be everything you want in a real small package, or something second hand on ebay thats been looked after as a full kit with everything you need already done
Oh and one more thing id swear by if you are networking things and sharing files between them would be a GB network card on both systems and hardwire them together, much faster than wifi, if they can take it, it makes file transfers so much faster and they are pretty cheap (im talking the towers here not much you can do with the laptop i dont think) you might need better cables depending what you are running but that will really improve on your overall experience, you can see how it starts snowballing and adding up fast lol you start poking at the SSD and before long you replaced everything
Whatever you decide good luck, i think you have all the info you need, its just what is going to work in your position. Sorry if i repeat myself, but id rather say things twice and make it clear than forget something critical