According to Article III, Section 16 of the Hawaii State Constitution:
The governor shall have forty-five days, after the adjournment of the legislature sine die, to consider bills presented to the governor less than ten days before such adjournment, or presented after adjournment, and any such bill shall become law on the forty-fifth day, unless the governor by proclamation shall have given ten days' notice to the legislature that the governor plans to return such bill with the governor's objections on that day.
The 45th day is Tuesday, July 8th, 2008. The Governor's ten day notification of intent to veto is Monday, June 23rd. The computation of the 45 day period excludes Saturdays, Sundays, and holidays.