You could possibly get as much as $15,000 but that’s unusual. My D17 was offered that amount with an ACT of 34 and an UW GPA of about 3.9 (honors and AP courses, etc.) but she was also a National Merit Semifinalist so eligible for the $10,000 Gold Award. She was admitted to Carlson honors but will be attending another school. We are in-state so obviously her tuition was covered via those awards, but not her room & board. She is was not eligible for any need-based aid.
If you make the honors program then you have a good shot at getting some merit aid. Make sure your application is in early on because that will increase the likelihood that you’ll have at least some idea of your award package well before you need to make a decision (assuming they stick to the same process as in previous years). All awards will be doled out by spring 2018 and the need-based portion of the financial aid package is also released about that time.
Remember, you can also borrow up to about $5,500 - $8,500 per year (increases every year) in Stafford loans. Even the unsubsidized rate is still pretty good. You won’t need to repay till you graduate and if you go right on to grad school it will be deferred further (although keep in mind that interest accrues throughout the life of the loan, even when you are an undergrad, and is added to the principal). Stick to about $20,000 in debt or less so that you have a managable amount to pay back. See if your grandmother would agree to make interest payments while you are in school - at least for a little while - to keep the principal from increasing. Also, make sure you and she understand how those interest payments work because they are going to increase big-time as more loans are added on.