REAR WHEEL DRIVE! Thats my selling point, but thats just me...
Not sure wich gen eclipse your looking at or wether it's a GSX or GST or NA?
Rear wheel drive cars are easier to work on (my .02), they drive better (again my .02), they slide better (my .02, you getting the point yet?), the miata is light, and most wont expect you to seriously race a miata (they'll be busy making fag jokes and talking about your penis)

Both the eclipse and the miata have large aftermarket pools to pull from. Im not sure if the eclipse does any circuit style racing or not? There are tons and tons of miata clubs, series, parts, classes for racing, the list goes on and on.

Performance wise, i know that the 4g63t engine is a frickin beast!!!!!! I know some guys are building the miata engines and getting some decent numbers, however some are hiding 302's and 350's under the hood (go ahead talk shit with your weedwacker )

Regardless of which you come across first, they'd both be good starter projects (providing money is not an issue), although i would prefer the miata over a fwd gst! A gsx however, might make me think about it for a few minutes....