There are ways to do this, but attaching it to the report system is not one of them; Riot would just end up with a bunch of people reporting everyone that didn't perform good enough for them in an attempt to only get "good" allies. While false reports are thrown out, it still would reduce the effectiveness of the report system.
Currently, the main reason players see someone twice in a row is because they both immediately requeued at the same time (since they are already in the same MMR range, if you queue at the same time you are going to be considered for a match together). If you don't want to see them, the solution is to just wait to requeue.
That aside, a blacklisting solution not tied to reporting could be to simply add a blacklisting feature to your "Recently Played" list. Since a limitless and permanent blacklist feature is abusable (especially by highest elo), they could limit the blacklist to only a few players, and limit the time they are on the list as well.
In the end, it's a bad idea to tell people to report those they don't want to play with, because that's exactly what will happen. You don't have to commit bannable offenses for someone to not want to be on your team.