If playing a custom game as a standard empire, use 'starting' empire expansion and any tech level other than 'prewarp.' That will give you your homeworld, a spaceport, a small number of warp-capable vessels, and a few mines (some of which may be in nearby systems rather than the home system). Each expansion level beyond 'starting' gives you a couple more colonies as long as your tech level is not prewarp; if your tech level is prewarp, then you always start with your homeworld and nothing else.
Also note that you can set the empire tech/expansion levels and the game tech/expansion levels separately (the game levels are on the first page of custom game setup, the empire levels are a couple pages on from there), and that if you set the game tech level to prewarp and your empire tech level to anything that starts you with a warp drive while staying at 'starting' expansion, it seems as though you're less likely to start with out of system mines and generally have less stuff, though I'd warn that starting with warp tech when everyone else is in prewarp is an incredible advantage.
If playing a game using the presets, I'd expect that any of the options other than Ancient Galaxy and Age of Shadows would start you with your homeworld, a spaceport, a handful of warp-capable vessels, and a few mines, though as the only preset I've played with is the Ancient Galaxy I could not say for certain.
< Message edited by Aeson -- 1/10/2016 4:37:12 PM >