The best card from Nvidia for that price range is a 1 gig 9600gt. I found one like that complete with an HDMI port. It's MSI brand so it's fairly high quality construction. I've purchased a lot of cards and a solid motherboard from MSI, and they haven't let me down. Final cost is $92.99 with free shipping, and you could even send in a $10 rebate if you wish to bring the cost down to $82.99.
Newegg.com - NVIDIA N9600GT-MD1G GeForce 9600 GT 1GB 256-bit GDDR3 PCI Express 2.0 x16 HDCP Ready SLI Supported Video Card - Desktop Graphics / Video Cards
The unfortunate thing is graphics cards are never shipped anywhere with an HDMI cable. They only occasionally have adapters for DVI to HDMI conversion. You'll have to buy the HDMI cable separately if you don't already have one.
The only card better than the 9600 from Nvidia is the 9800GT. You may find a good 9800GT lying around at a similar price, but almost all of the ones I've seen were about $120 dollars and up. So unless you feel like shelling out the added cash, which I doubt, the 9600GT should do you just fine.
If you're a Radeon fan, the best you could go for is a single Radeon 4850 512mb. (They also come in twos in a crossfire configuration.) If you really want big bang for your buck at that price range, this is probably the better choice. It'll cost you $94.99 with free shipping. The best part is that the 4850 is comparable to the 9800GTX in performance. So it's about 40% more powerful than the 9600GT at about the same price.
The only downside is this model doesn't come with an HDMI port built in. It comes with an adapter though.
Newegg.com - SAPPHIRE 100245L Radeon HD 4850 512MB 256-bit GDDR3 PCI Express 2.0 x16 HDCP Ready CrossFire Supported Video Card - Desktop Graphics / Video Cards
You probably won't find much else better than this at that price range. Hope this helps.