Since the creation of eGPU adapters, it seems that there may be a possible way (even though it is somewhat ridiculous) to improve the graphics capability of the Dell Mini. However, the only model that seems as if it would be easiest to do with is the Mini 1012 - 3G/Broadcom Hd model, with the extra 2 mPCIe slots in addition to the 3rd one for Wifi.
The reason for this is obvious, you can keep your Wifi, while using the eGPU. Also, the orientation of the mPCIe slot makes it perfect so that you could actually run the HDMI cable straight through the side of the case. It may even fit inside hole for the card reader, without actually needing to make huge modifications to the bottom chassis for an additional hole. However, you could also just cut a hole and make a dedicated HDMI port off the left side (between the card reader and the USB) so that the eGPU can be unplugged when not needed.
Most of the eGPU projects I've seen have had an external enclosure for the graphics card + eGPU adapter. Most of them are also very crude and ugly. I was thinking the best route would be to make a netbook stand, or sort of like more of a docking station. The eGPU would be mounted into it horizontally, so that the ports on the graphics card are facing to the back. It may be worth gutting out an old computer case and re-purposing the PCI brackets for the netbook stand. You could also make this netbook stand into a cooler for the bottom of the netbook, with the addition of a USB hub, and even integrated speakers? I think the most power efficient + price/performance GPU would be the GTX 750. But I mean you could go lower in terms of power, but the efficiency just isn't there. Though, the GTX 750 would be bottle necking the Intel Atom CPU like crazy, but there's not much that can be done for that.
I'm not sure what type of material would be best to construct this, but I'm thinking acrylic sheets, or OSB hardboard sheets. I think plywood would be a bit overkill. And I'm not much of a machinist so sheet metal would be out of the question.
Again, this is just food for thought. I have no plans to attempt this project, but its interesting to imagine the possibilities of doing so. I also think it would be somewhat cost prohibitive, considering the amount you could spend on a half-decent gaming laptop in today's market. You would need to find a way to prevent it from being too bulky, otherwise, it loses purpose.