I would like to ensure maximum quality of my images after downsampling. The downsampled resolution will be 300 DPI for print. I need the source graphics to be ever so slightly larger for use in an app. I can pick the source resolution ahead of time, so I'm wondering if some numbers will yield better results when downsampling? 320, 352, 360, etc.
It really all depends upon the image itself. And the software perhaps. You don't mention what software you may be using.
The thing to keep in mind, with Photoshop at least, is that interpolation of data happens with any resampling. So if you need to drastically resample an image the greater the steps you do that in, the more interpolation occurs.
The theory is to to it in 2 or 3 steps. That allows for more targeted interpolation in stages rather than everything at once.
So, I generally down sample in 2 steps... one to 50% of the necessary down sample, and then again for the remaining 50%. If the image merits that. i.e. a 360ppi image doesn't need 2 steps to down sample to 300ppi. It's such a small jump that 1 step is fine. But for a 600ppi image, I'd down sample to 450ppi, then to 300ppi.
And all this is even if down sampling is necessary. Much of the time, any image above 300ppi will be down sampled by PDF job options or output option. So, direct down sampling of the image isn't really necessary. The only time is may be necessary is when there are critical details in the image one is trying to ensure are preserved as much as possible. And in that case, it's really on an image by image basis. Some images may require 1 step, others 10.