I really would not want to go below I have shown, but this card GV-N630-1GI NVIDIA GeForce GT 630 1024MB DDR3 PCIe 2.0 x16 is still falls within entry level gaming card
http://www.microcenter.com/product/391178/GV-N630-1GI_NVIDIA_GeForce_GT_630_1024MB_DDR3_PCIe_20_x16_Video_Card and, despite recommended wattage, it needs only 65 watt to work, so no power supply upgrade needed.
To find cheaper and still able to play today games in tolerable parameters would be more difficult. GT 630 will let you play BF3 somehow.
I really would not want to go below. This card also was reported having very good cooling abilities, which is important for your case.
I would like to stop here.
If you want even cheaper, go to Microcenter site and look around, then provide links here, so I can check if it will fit.
Actually I will provide guide, which can help you choose from the cards the one which will fit, this is how I look for proper fit in Optiplex Mini Tower.
THIS POST APPLICABLE TO MT (MINI TOWER) FORM FACTOR ONLY.
Unfortunately no card that is more than 7.5 inches long or 2 slots wide will PHYSICALLY FIT inside the case - it was found that only single bracket video card can fit due to BTX MOBO, where card internals facing upward towards PC internals, not ATX, where dual card design is blocking another PCI slot by facing downward.
If you prefer the card with beefier fan, then the length of the card must not exceed 7.5 inches, otherwise only "skinny" single bracket, single slot cards can physically fit.
Pictures below will prove my point.
View shows no room for dual bracket
Single bracket only, or you have to cut dual bracket in half
This is the other way to show my point
Here you can clearly see only single place for the bracket
Visual explanation
7.5 inches explained, also shown "obstacles"
This is the picture of actual "skinny" single slot video card (not sure which model)- perfect, "glove" fit.
Good luck.