Realistically any new GPU will push video fine at 1080p. A budget hd6450 is probably only $20-40, but its not going to give you much headroom for anything else. There are plenty of those that are passively cooled. As soon as you want something better than that, that is passive, you are looking at a premium.
Newegg has a passive hd 6670 with ddr5 for $85.
Newegg.ca - SAPPHIRE Ultimate Radeon HD 6670 1GB 128-bit GDDR5 PCI Express 2.0 x16 HDCP Ready Video Card (100326UL)
You can get actively cooled (but still quiet) hd 6670 for ~$60 after MIR.
If you want up to date passively cooled card this is the best deal I could find (HD7750 passive for $100 and a $10 MIR to make it $90).
HIS Radeon HD 7750 iSilence 5 1GB PCI-E w/ Passive Cooler, DVI, HDMI, DisplayPort at Memory Express