The precipitation and temperature data you prepared for your site are entered by giving ArcSWAT a file of station locations; this file also contains the file names of the files that actually hold the P and T data -- as you probably know.
The "database" referred to here, I believe, is the table containing the locations and statistical parameters describing typical weather at a number of locations in the USA. These data are used to generate missing weather data that SWAT may need. Commonly these are for wind, relative humidity, and solar radiation, data needed for some ET calculations. To create your own custom database, you would need the locations and statistical parameters of weather stations around your site. I'm sure there are programs that can translate long-term climate records into these parameters, but I haven't used them.
The Hargreaves ET method uses only min and max T, I think -- perhaps if you choose this method, you may not need a weather database -- or if you choose the US database it won't matter, because SWAT won't need any extra weather data that you're not already supplying. I am not sure of this; perhaps someone else on the list-serv can confirm or correct this possibility.
Good luck,
-- Jim