Your in state flagship would be an excellent place to start.
If you are in a western state, you can get reduced reciprocal rates at other states in the west (minus TX). However, if you are in CA, WA, or CO, your best bet for value/prestige are going to be your instate schools. Apply early for best consideration.
http://wue.wiche.edu/search_results.jsp?searchType=all
There is good (but not guaranteed merit) for high stats kids at U of Arizona, Arizona State, U of New Mexico, and UT-Dallas that will probably get you within range of your budget. I know high stats students at all of these, who followed the money, and are having great experiences. They increasingly are attracting a lot of students from CA, because the cost can be less than CA publics for high stats kids.
Other places that are strong for merit are U Alabama (Tuscaloosa or Huntsville) and U of Miss.
If she wants to try for the brass ring and apply for (very) competitive merit based scholarships by separate application that go up to full tuition, try Wash U, Vanderbilt, Duke, UNC, UVA (?), U of Richmond (private), Case Western. Some of these have STEM-specific scholarships.
These scholarships (also long shots) emphasize community service and leadership.
http://www.stampsfoundation.org/partners/
Women’s colleges can also offer good merit - Mt Holyoke, for example. Most LACs that offer merit scholarships, however, won’t be enough for your budget. Typically, you might get a maximum of 1/2 of total cost discounted, which will still leave 30-35K left over to pay per year. Some exceptions are Denison (OH), Lafayette (Marquis Scholarship), Muhlenberg ¶. I think that Rhodes, Centre, and Hendrix also have a few larger scholarships, but I don’t remember the details.
If your child is interested in engineering, most LACs won’t support it except in 3+2 programs, which I wouldn’t recommend. Other science/math concentrations can be very well supported in a smaller environment.
My advice - lock in an affordable safety as early as possible (auto merit for stats) that your child would be happy to attend. Then try for the long shots.
Best of luck to you! The good news is that it is possible. There are many excellent schools in the US. Your child may end up at one that you have never heard of before, but will nonetheless get a superb education and a memorable 4 years.