Basically it is each school that defines what their honors courses are,
and the better students take them. Honors students are more enthusiastic
about their courses, going beyond bare minimum learning requirements. A
good home school has the capacity to have all students doing honors
courses, because it would encourage their students to go the extra mile in
their studies. However, when a home school declares their students to be
honors students, that statement rests on the integrity and credibility of
the home school itself. There are some colleges that recognize that home
school students are some of the better students, and so the name of home
schooling has grown, but this is not guaranteed. You may want to consider
certifying your students with other certificates such as CLEP.
CLEP is a good way for home school students to earn credits for college
based upon what they already know.
Advanced Placement courses with the college board are another way to do
undergraduate courses as a high school student:
Apologia's opinion of their own courses
In the description of each book, Apologia state whether the course is college prep or not. See their list of Jr and Sr high books.
See their knowledge base article 27143 "Are our courses college-prep or AP?"