If you're a California home "owner" then the need-based aid is probably not for you unless you have 2-3 kids in college at the same time, and even then maybe not. You know that you need to fill out the FAFSA, probably online, and that it'll give you a number for "expected family contribution" iirc. For a California homeowner, that expected contribution may be something like 50K$/year.
So how about non-need-based aid? Some schools Just Say No. St John's College (Santa Fe or Annapolis), Carleton College, and I think Macalester, as well as other name-brand schools, fall into this class. What's that about, anyway?
Here's my understanding of it: colleges don't do non-need-based aid for love; they do it because they want their student body to have certain characteristics. St John's, Carleton, and some of these other schools can just admit the kids they want, because a high percentage of admittees actually attend.
Other schools use their financial resources to "engineer" their incoming class. Schools seek racial diversity, geographical distribution, gender parity, and so on; so if you're looking for an incentive, find a school that wants what you've got.
How do you find such a school? Fill out the profile on the PSAT (do they still call it PSAT/NMSQT?), and if you're in an underrepresented class, then for goodness's sake let them know about it! Then watch the marketing literature come in. Some of the schools that try to "sell" themselves will have incentives available.
It goes without saying that you should try to get a good score on the PSAT and SAT. National merit finalists have doors opened for them and money thrown at them. It is not necessarily a lot of money, but it could be. Rumor has it that a national merit finalist can get a full ride at the University of Fairbanks for example.
A word about underrepresented classes: if you're applying for admission to the University of California and you have a Japanese or Chinese surname, you're not in an underrepresented class. BUT if you're thinking about going to Calvin College in Grand Rapids (not a bad choice in my view) then you would be underrepresented there, and Calvin has some incentives to offer. Other schools do, too, but I know about Calvin 'cause I've visited there.
Unless you're headed for a traditionally mostly-male school, then boys are underrepresented in colleges these days. So it usually helps to be a boy, if you're looking for financial incentives.
Sometimes a visit will help. I know one father who visited Oberlin with his daughter. In the last interview of their visit, he said "she likes the school a lot, but I'm looking at an additional $80,000 over 4 years, vs. sending her to UC Santa Cruz." The admissions officer said something like, "Well, let's see if we can't do something about that" and basically made up the difference!
I have also read of colleges whose staff reasoned, "they came to visit, so they must be motivated to attend here -- no need to bribe them" -- in other words, visiting reduced their chances of getting financial aid. (If memory serves, that particular college has changed its ways.)
Pure dumb luck plays a role here. There was a college that was trying to increase the number of out-of-state students (as distinct from international students). In particular, one admissions officer had the mission to double the number of California freshmen over a 2-3 year period. Financial incentives were available, up to and including a full ride scholarship to this highly-regarded public university on the east coast.
This sort of thing is quite rare; I expect that the more usual thing is for a college to offer a nontrivial incentive ("scholarship" or "bribe" or "discount" if you prefer) if they think you might come, and pay the rest. Here's what I mean. Suppose two colleges, each charging about 50K$/year, compete for similar students. Now if college A offers a 12K$/year incentive to a student, and college B doesn't, there's a fair chance that college A will collect 38K$ from that student's parents, whereas college B collects zero. Obviously college A can't offer that big an incentive to everybody, but if you're the kind of student that fits college A's desiderata, that could help your cause.
So here's the summary of my advice: Study diligently but not obsessively, get the best grades and best test scores that you reasonably can, read the marketing literature, take care in filling out the applications, be earnest in the essays, and pray both for guidance and provision.
NOTE: Sorry, I started off addressing this to parents, but then ended up writing as though to students. Maybe I'll fix this later...