I too am having this problem. No matter how many times I try to "solve" the alert issue, it keeps on coming back. I can't say exactly what started this rash of annoyance, as I updated the printer's firmware and I switched to Leopard (OS X 10.5) last week.
Also of curious note, when I click "Solve" and the HP Printer Utility starts up automatically, the Color Center portion is lacking the ICC calibration and Manage Papers menu options. But if I start the Printer Utility on my own, those items are there. Also of interesting note, if I have the Printer Utility open (and those ICC, calibration, and paper items are listed), and that alert pops up... hit Solve, and focus switches to the Printer Utility, but those Color Center options magically disappear, while it's open. Strange!
-J