I think the following would be a clear improvement over the current draft system if you're playing with a full pre-made team. This would only apply to games where your entire team is pre-made.

Bans work as normal, with Blue getting a ban, then Purple, and so on. Then any one person on Blue locks in a champion. It doesn't have to be the top person, just whoever selects and locks it in first. Then any two people on Purple lock in their champions, and so on. This way, people still have to own the champion they're going to play, but your team can be perfectly flexible with their picking order.

This would greatly help new teams who are looking to climb the ranked 5's ladder but don't have the time or money to have gotten all 100-something champions for every member of their team. As it is, when I queue with a full pre-made group, we are often forced to pick our champs in a highly non-optimal order, and we sometimes miss securing priority picks because the person who owns that champ isn't up next, and so on.

Here's a variant that could work for both solo queue and pre-arranged games. If you haven't yet locked in a champion, and you're not in the last round of picks for your team, you may click a button to offer to trade pick orders with one of the players further down the list. (I'm visualizing it as having the same interface and mechanics as trading champions. So you click a button by their name and they have to click "accept".)

For example, I'm first pick on Blue side, but I'm top lane and I don't want to pick right away. Me and our jungler agree to swap pick orders, so I click the button and we trade spots and he locks in his choice. He used to be pick #3, so I am moved there. If in the next round of picks, I still don't want to lock in my champion, I may be able to persuade either the 4th or 5th pick to swap with me.

The one downside to this suggestion is that sometimes people who swap away their higher spot may be trolled. For instance, in my example, the person who said they jungled might decide to steal top after swapping with me. Or maybe somebody else who I was moved behind steals top. However, either of these things could happen with the current champion trading system, where I pick a champion for somebody and expect to get a trade for what I want after the picks are done.

However, this swapping pick order idea avoids one big problem with the current champion trading system. Let's say I'm playing with the current system and I'm first pick and Blitzcrank is open. Our 4th pick says "get me blitz!" and I lock it in. Then later on I find out that the 4th pick doesn't own any champions I want to play (this can happen even if you check beforehand, if the other team gets the champ you checked for first). This awkward situation would be completely avoided by my suggestion, so I think overall this would be a positive change.