So this is interesting, because I do fantasy racing in another sport (Motocross and Supercross) and they have had a hell of a time figuring out something that sticks. A pitfall that they generally run into is that you just pick who finishes 1-5 and then a random position between 7, 9, 11, 13. This is fun for about 3 weeks, and then you're picking the same guys repeatedly. Some of the better fantasy options have gotten around that by doing what I'll call "handicap racing". Lower tier guys who usually finish outside the top 10 are given a position handicap; so if a guy has a +8 handicap and finishes 11, he pays out points equal to 3rd. The handicaps are dynamic and change week to week; on what, I'm not sure of the specifics, but I imagine combo of most recent 2 or 3 finishes and current championship standings. The kicker is that, if the handicap rider finishes inside the top 10 after his handicap, you score double points, so value is there in picking the lesser known riders. Supercross in particular has a fun qualifying system; there are usually about 50 or 60 riders trying to qualify at any given event. Only the top 40 in qualifying make the "night show", which then pares down even further into the top 22 for the main, based on heat races. 20 guys each, top 9 advance. Remaining 24 after that go to a Last Chance Qualifier, and the top 4 make the main. So there is a real possibility a guy you pick with a high handicap doesn't even make the main; picks are due between qualifying (~3pm EST) and the start of the night show (7pm EST).
What the handicap racing has done, at least in this context, has made the racing in the middle/back of the pack stupidly fun and entertaining for us when we play. For F1, everyone already watches the leaders, and the manufacturers at the front are still probably going to be the top 4. But the intrigue now becomes what happens in the 10th-16th range, where stuff really gets mixed up. Due to the handicaps, one position could be the difference between like...12 points and 36 points. Big swings off of passes that probably don't even make the broadcast. It also really helps carry the less entertaining races too.
All that really just to offer you some insight on what could be done. Fantasy Racing is something I really like, obviously, and while you may not be in it to make a premium product, or rake in ad revenue or something, this could help set you apart and add some additional fun to the setup over the traditional "rank the drivers 1-10 based on qualifying".
I've been seeing more and more "homegrown" fantasy stuff pop up recently and I love it. I signed up, will try to remember to make lineups. I'm not a huge F1 fan, I just love following racing, but I'm trying to get more into it. It's just difficult with the start times being all over the place!