The jQuery time-picker plugin that I wrote uses a div as the containing block for the list of times, and on Mobile Safari there are no scrollbars to indicate that there are
Mobile safari, as far as I have seen won't support scrollbars. The best plugin I could find to get the job done is this.
Its Demos are available here. It also has multiple predefined skins to suit your application.
here's a sample of what you'll get -