I asked about the motherboard because sometimes the BIOS won't accept certain CPUs, but according to this http://www.findlaptopdriver.com/hewlett-packard-304bh-mainboard-cpu-support-list/ it should support the i7-880. If it doesn't support it natively, then it should after a BIOS update.
Here comes the consideration part: something like the GTX 1060 has a 400W PSU requirement.
The least power hungry card I know about is the GTX 750 ti as it doesn't require additional power cables. Plus, the size of the case is something which has to be considered and it seems to mean getting a low profile graphics card. This would immediately effect what you could expect from your games. If we stick to low profile cards for gaming, then only the GTX 750 ti and the RX 460 are contenders. There are models for both which are low profile and run off the PCI slot without additional power connectors.
Whether this will meet your gaming expectations is questionable. Ideally, transferring the motherboard to a new case (assuming all the mounting points conform to industry standards) would then give you far more upgrade options including PSU and GPU.
Anecdote: when I upgraded my pre-built study PC, the graphics card I bought was a bit too big. I had to buy a new case. Luckily the motherboard did conform to industry standards so transferred easily. I had to get a new PSU later as the old one popped (probably due to power demands). So be careful in your choice of components and how you go about the upgrade.