[brianogilvie]

I don't have an iPhone, so my comments refer only to the desktop version (1.1 sneakypeek).

In the Data Preferences, you can set the default time for due dates. If your work day ends at 5 p.m., make that your default due date if you don't want things due today to be overdue.

More generally, the GTD method that underlies OmniFocus's design frowns on using due dates to indicate things you simply wish to get done on a certain date. Instead, due dates should be used only for actual deadlines. Many of us use start dates as a kind of tickler; by setting the start date on a project or action, you can tell OmniFocus not to bother you with that thing until the day arrives. I've set up a Tickler perspective that shows me actions whose start date is in the next week; I use that at the end of the day to take a gander at what will be on my plate soon.

If you're not using flags for something else, why not use them for things you want to do today? In the morning, scan your list of available actions and flag those you want to do today. I'm told that the iPhone version of OmniFocus can show only flagged items.