Once it starts flashing that means you're either at or near the service interval (7500 miles for the '01), it flashes for 10 seconds for a fixed number of miles then stays on solid after that. The older Hondas had the little green/yellow/red light, this is just a different version of the same thing.
I haven't heard of the switch in the glovebox. However there is a simple method to reset it. I don't remember all the details, but it's something like hold down the trip reset button and put the key in the ignition and set it to run, keep holding for 2 seconds then release. If you want to reset it I can get you the specific instructions out of the Helm's manual.
The Owner's Manual of your car will explain the different mileage limits for the blinking vs. solid.
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