Do not use a large zoom lens. They compromise image quality across their large ranges, and you don't need large zoom ranges for product photography (unless you are selling ants). Go for large sensors, small pixel counts, a tripod and a lens that is good around 50mm EFL (effective focal length compared to 135 "full-frame" format).
A flash hotshoe is mandatory.
Personally, I like my DSC-R1: it has an APS-C-ish sensor (crop factor 1.66), just 10MP, base ISO of 160 (and one shouldn't go all that much higher because its sensitivity as one of the first CMOS cameras is not fabulous) and a very good fixed zoom lens with 24mm-120mm EFL with f/2.4 to f/4.8. Its main disadvantages are that its LCD and viewfinder resolution are not enough for evaluating picture quality, and its JPEG processing is definitely inferior to working with its raw files (which are a full 20MB in size and cause some delays with the old media format this camera uses).
There is some merit for product photography of small items in getting the VCL-M3367 closeup lens (+3 dioptres) which allows getting significantly closer.
The optics of this camera are impressive at its price point and will give kit lenses a run for their money, but newer cameras will offer faster handling, displays with significantly better resolution, more sensitive sensors (not all that important since product photography is not done in low-light), better JPEG processing.
And make no mistake: you'll likely have to invest a significant amount of money in lightboxes, flashes, softboxes and other light modifiers, remote flash triggers and whatnot. So the price tag of the camera will likely not end up taking the majority of the budget if you are pitching for older gear.