I'm assuming it's a slow HDD, not a 7200+ or anything? You might want to consider one of those hybrid drives since he won't see the full speed benefits of an SSD.
A good 7200 rpm HDD, i.e. 250GB or more would bring a meaningful improvement as well. Even a rather recent 500GB 5400 rpm laptop drive is rather fast (console hard drives)
I'll be way slower than SSD, but useful if it doesn't cost anything, free of worries about TRIM or possible issues with a low end, small SSD.
But for video consumption, youtube-ing and somewhat frequent web browsing, the CPU will still feel slow I imagine. And the ssd won't help here. So perhaps a cheap AM3 upgrade is in order.
Edit : wrong about AM3. If you switch mb you'd need something with a modern IGP to offload some video decoding. So it's either FM2 slots for 2M4C AMD APU or some Intel 1151.
On Intel socket you'd get only (Celeron & Pentium) dual cores for the cheap. I've used a Skylake 3Ghz Pentium for 6 months, it's a terrible experience. So I'd recommend at least higher clocks for this choice
It's 2017 and I make do with unaccelerated web video. It's a bit painful sometimes (linux with open source radeon driver)
With a fast CPU, performance still seems to depend on the graphics hardware even though anything from ATI Rage Pro onwards should be quite good for raw 2D in theory.. E.g. I see a Core 2 Duo E7500 with Intel graphics (G43 or something) working quite well, but a Q6600 with geforce 6100 (called 7100 there) seems to struggle with HTML5 video. That's weird.
Anyway, I wonder of the pertinence of getting a geforce GT610 or 710 for a 3GHz Pentium 4. If it has PCIe that is. Should work with even 1080p 60fps youtube, etc.
You get a super fast and stable graphics driver as well, reclaim a bit of memory capacity and performance if you were using integrated graphics.
My educated guess is software rendering of HTML5 video does stupid things, like software YUV to RGB conversion and software scaling, then pasting playback controls etc. on top. This hammers the CPU memory bandwith, and then the integrated graphics reads the result in system RAM to display it. So, perhaps a Pentium 4 with old integrated graphics or an old netbook will be able to read 720p H264 in VLC, but it will fail at full screen 360p youtube.
If you have a Pentium 4 with good RAM and a Radeon 9200 or such, perhaps it does 360 or 480p fine?
Web browser : try ublock origin (crap blocker with most efficient CPU/RAM usage), Firefox is good on low memory PCs too.