Son had same dilemna with UofAZ and a host of top schools IL, WA, WI. He chose the free route and has not regretted it at all. Be aware that, as an engineering major, they can struggle to complete the Honors college requirements since most of them are not STEM related and the Eng degree program is intensive in those fields (at least at UA). Additionally, if they have a lot of AP credits, they will pass out of many Honors eligible classes.
As an incentive, we had him take out the max loan amount for students (varies by year, but roughly 6-7K) to help cover outside expenses with the caveat that if he graduates with distinction (3.5+) we will cover those loans for him. Made him very aware of the costs of school and the importance of keeping grades up.