Here is a site that provides a good overview of public honors colleges: https://publicuniversityhonors.com/new-top-programs-by-category/
The “intends engineering but maybe business” suggests that you may want to look at the strength and offerings of both the engineering and business schools. Anecdotally, engineering is starting to sound to me like pre-med, meaning that smart kids who are good in STEM choose it as their major initially, and then a chunk decide that it’s not for them.
When I took my D to visit colleges back in 2013-14, she was focused on engineering, as were the majority of her smart, high achieving friends. A number of them chose public Honors Colleges or programs. Fast forward four years (she just graduated) and only one of the bunch who intended to major in engineering actually did so:
D went to South Carolina, a highly-ranked public honors college, She majored in International Business (one of that school’s strengths) and Econ; has a very good job, zero debt, living on her own. Will start company-funded CFA process shortly.
Penn State (Schreyer Honors, also highly ranked): Dropped engineering, transferred to Wharton in Year 2, majored in something business-y and works for a boutique investment bank in NYC.
U Vermont Honors: Switched from civil/environmental engineering to some sort of environmental studies major and is now at her parents, working part time and prepping for the LSAT.
Two or three friends went to Pitt for engineering (don’t know how many were in Honors, but Pitt’s Honors program is not so structured as some others anyway). They all switched majors within Year 1 (some within Month 1!), and graduated on time (with jobs, though I don’t know what they do).
U Delaware Honors: Switched from chemical engineering to Accounting almost immediately and works for a Big 4 Firm that’s paying for her grad work in Data Analytics.
Temple Honors: Entered undecided, considering engineering; ended up in Actuarial Science with a very good job.
Rice: Started in engineering but switched to something in business pretty early on and now works for a hedge fund in NYC – just to show it wasn’t just the public school kids who switched out of engineering.
The only one from D’s group who stuck with engineering went to Stevens and loves it. She will graduate in five years with some lucrative summer and semester-long co-ops with good companies. She will likely be highly employable even though she’s in biomedical engineering.
Obviously this is just one person’s personal experience. I don’t doubt that a number of D’s classmates stayed with engineering and did well; they’re just not the ones she kept in touch with.
My point is that not every kid, even the very bright ones, who think they want engineering will stay with it. Large schools like UIUC, Purdue, Alabama, etc., have different strengths beyond their engineering departments that may be worth investigating.