In that order. Depends on the Motherboard, you could try to upgrade the CPU (it might be cheaper since it is older stuff) but I would upgrade RAM first to at least 2GB (really really cheap for DDR2 like 20$ per gig at Micro Center) then video card. You can get a decent card under 100$ for streaming. If it doesn't work, you can always return it
Surprise, it's likely going to be more expensive because it's that old, unless you get used from someone selling on eBay or Craigslist or Goodwill. The only stores selling those parts at this point are selling a decreasing supply of new parts to businesses that have need of keeping older computers working for some reason or another.
Second, adding more RAM won't help much. My last system was an Athlon XP 1.4Ghz running WinXP and it didn't see a lot of benefit from RAM beyond 512MB. The extra RAM is not going to help stream Youtube videos, because Flash is horribly inefficient at video playback.
A video-card might help, if your motherboard actually supports AGP, you can find a decent AGP card that will work with your system and that card supports hardware video acceleration. All three of those are iffy. If you've got onboard video right now, your motherboard may not have AGP at all, and even if it does it may not actually work (many motherboards from the time had trouble with conflicts between onboard video and discreet cards).
EDIT: It's going to be far more effective to buy a new system, and most likely it'll be cheaper as well. Certainly cheaper in the long run, because it's not going to be long before components of that relic are going to start failing from leaking capacitors and what-not.
EDIT 2: What are they wanting to use a computer for? One of the newer "nettops" might work for them. A dual-core Atom paired with Ion2 graphics would be surprisingly capable at most basic task, and really excell at most video playback.