Also, on the actual webpage, I built assuming for a width of 320px and had the following meta tags:
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<meta name = "format-detection" content = "telephone=no"/>
<meta name="viewport" content="user-scalable=no, initial-scale=1, maximum-scale=1, minimum-scale=1, width=device-width" />
<meta name="apple-mobile-web-app-capable" content="yes" />
So that's limited because it only works in iPhone portrait mode, but still, for a quick and dirty demo, not bad.
(For a quicker and dirtier demo of the idea: just add the webpage to the homescreen. When they start it up it will be in fullscreen, no-safari-random-widgets, and to the unaided eye looks about the same, except the icon will be a screenshot rather than something you can set.)
UPDATE: My buddy Jonathan Zaleski says
You can control the home-screen icon w/ the following:
<link href="favicon.ico" rel="apple-touch-icon">
Good to know!