This reply may be a little late for your graphic card upgrade party, but my experiences with upgrading Elite- and dc-series Convertable Midi Tower (CMT) Small Form Factor (SFF) desktops could be useful to other forum visiters as well.
Non of these basic HP desktops allows a mid or high-end Graphic Card, unless you'll replace the original power supply with a beefier unit. A rule of thumb with regards to power issues: almost any card without a 6- or 8-pin power supply connector will do.
My preference for graphic cards for sytems up to the Elite 8000/8100/8200 would be an ATI Radeon HD4550/5450/6450/7450/r240 or similar. Most nVidia cards in this price range (GT-.10 to .45 series) do not match the performance of these AMD cards, unless you're willing to spend some money on a Quadro .10 series (410, 610).
Some folks have succesfully implemented nVidia GTX-650Ti/750Ti (and similar cards) in their Elite desktops, but if you choose to go this route you have to make sure the bios of these cards won't cause UEFI related conflicts with the bios of your system.
I have installed a AMD HD6450 in a old dc5800 SFF with 8gb Ram and e8400 processor and the performance of this little pc is quite amazing. I have done some CAD modelling (Solidworks 2016), Photoshop, played some older games at Full HD etc. without major performance lags.
My mother uses an Elite 8000 CMT with AMD HD 5450 and 8Gb RAM; which is almost overkill ;-)
We're talking at decent computing on a very tight budget here. The dc5800 was $30, including the upgrade. The Elite 8000 was $45.
My advise for older Elite desktop owners would be to buy a decent (used) graphic card and put as much RAM in it as your system/wallet allows you to.
One last obvious performance booster would be an SSD. For older (SATA-2) HP desktops a (used) HP approved Intel SSD would give you optimal performance. The more recent systems that support SATA-3 will take any SSD without hassle.
The performance of Intel GPU's in more recent Elite Desktops (6300, 8300, 800 G1, G2 etc.) makes an upgrade to a dedicated Graphic Card less useful.