There's really two options for a cheap 10-second car.
One is to go with a big-block V8. A 440 Mopar engine is the smallest I'd even think of considering for this, and the bigger you can start with, the cheaper it'll be. Chevy will be cheaper than Ford will be cheaper than Mopar, though I'd probably go Ford just because I know them better. Take that big cube engine, get a hotter top end on it (cams, intake, heads if you can afford it), and stuff it in the smallest car you can make it fit.
The other way to go is to start with a turbo 4-banger, and shove as much boost as you can through it. To go that route, you're pretty much looking at starting with a 4G63, 3S-GTE, or SR20.
In the US, the big-block route is definitely cheaper. The most expensive part of the build is the heads, and good-flowing BBC heads *might* be $1500 for a set of fully-assembled aluminum ones.
