Abuse. There isn't a reasonable enough system (that I've heard about) that can effectively let you leave toxic games, without being abusable.
If you got 3 free dodges every month? Well then they'd be gone in a day or two and you'd have the same problem. You got 3 free dodges every week? If you don't play a lot, this might be your ticket, but now EVERYBODY has a bunch of dodges. You know how many players out there will dodge because their teammate picks an off-meta champion, don't lie to yourself.
There is no real way to give only the good players the ability to dodge and not the bad players. You could give them out based on reports, sure. If someone gets reported, they don't get their weekly dodges. But then, if they are the type of player that you would report, you'd just wind up wasting a dodge on them every time anyway, wouldn't you? That person in of himself is like -1 dodge to anyone who gets matched with him.
In my opinion, 5 minutes is a small price to pay in order to get a second chance for a better team.