AS you already know, your current integrated Graphics card really cant cut the requirements to run FFXIV in a respectable way.
Good news... your CPU and RAM is good enough, and the standard spec on your Dell will mean the graphics card slot on the motherboard is empty.
Now you can pick up a decent pre-owned one off eBay from $50+, so I suggest looking for one with ideally 1GB Ram (go with a better brand name like Asus as they will be better made l have better support. No point looking at anything larger than 1GB as you wil probably need to upgrade your PSU, etc.

BTW it may not be easy to find a good enough spec machine for $250 unless you see one on sale (or on eBay) as pre-made machines/laptops run on shared memory graphics chipsets.

Hope that helps.

p.s. if you require more info or anything else just feel free to ask