Probably the biggest difference today is that integrated graphics are now powerful enough that you can run even pretty graphically rich titles at 1080p on modest settings. If I weren't pushing the envelope too much, right now I'd wait for someone to release a small-form-factor PC (like a Gigabyte Brix) based on the coming AMD Carrizo APU (should be any week now), drop a 64-128GB mSATA or M.2 drive in it, with 16GB of fast DDR3 (fast stuff because the GPU benefits measurably) and call it good. You'd have something
nearly as powerful as not terribly far behind the current-generation consoles, maybe half as powerful as Xbox One. If you needed a little more grunt, Zotac makes some small computers that have laptop-style discrete GPUs inside, those can easily match or surpass the current consoles.
Lots of the newer-style arcade machines are running on DVI/HDMI - style LCDs now, so you don't need to worry about the wonky refresh rates that the old arcade tubes used, and demanded specifically-tuned GPUs to drive them. If you opt for an authentic modern arcade LCD, this'll probably be the most expensive component, probably between $600 and $1200 depending on the size.
With a little more horse-power, or maybe for basic 2D games, you could probably even use one of those inexpensive Korean 34" 4k displays (about $400), though they'll only do 30hz at 4k and 60hz at 1080p.
Also in that time, ITX form factor has really been adopted widely, and you can build a very powerful system that way. My current gaming rig is an ITX boad with an i7 4770 CPU (water cooled even), 16GB RAM, and a Radeon 290x. The case is basically the same size as a 24-pack of soda cans.
On the complete opposite end, if your needs a really minimal, a 1080p display and a Raspeberry Pi 2 wouldn't be bad. Its got GPU that's somewhere between the original Xbox and the Xbox 360 (its capable of pushing games akin to the late Xbox/early Xbox 360, at 720p) and a quad-core 900Mhz ARM CPU with SIMD, and 1GB RAM for 35 bucks.