I am going to go back further. Almost twenty years in fact...
Video: Orange Road
Creator: Vlad G Pohnert
Series: Kimagure Orange Road
Year: 1993
Why Influential: September 1993! Full SD 640x480 captured from LDs. IN FRIKKEN 1993! If this isn't the birth of Anime Music Videos, it sure is close. Vlad may be modest, but given that I think the quality still holds up today, and it is still a very fun video, he had the goods 17 years ago, and there's no real way to judge just how much influence Vlad has had on the hobby because of early videos like this.
Video: Fields of Gold
Creator: C-Ko
Series: Naussica of the Valley of the Wind, Laputa
Year: 1994
Why Influential: See description for the next video below.
Video: My Euthanasia
Creator: Lorraine Savage
Series: Various
Year: 1996
Why Influential: This video, the previous one, and the next one are more about the era than these individual videos. Many of these videos are lost to us, as they existed on VHS because that is how they were made for some of the first contests produced. Deck to Deck. No transitions. Just straight switching from A to B. Those that survive help inform us about the hobby we are part of and should be cherished. Without them, we wouldn't be here. I had just gotten into anime fandom heavily in 1996, and it would be two years before I knew what AMVs were and got involved in them, but these are examples of some of the first videos I saw.
Video: 500 Miles
Creator: Duane Johnson
Series: Ranma 1/2
Year: 1996
Why Influential: See above (or below

)
You can see the entire list for the 1996 Otakon AMV Contest below (kinda gives you an idea of what it was like then).
Video: Engel
Creator: Kevin Caldwell
Series: Evangelion
Year: 1999
Why Influential: Do I really need to explain? The fact I
don't is the whole point of including Engel. Of all of the AMVs I know that just random people could name, Engel probably did more to spread AMVs to new demographics during the early days than any other AMV I could name. I've met a number of people over the years with only the slightest interest in anime, let alone AMVs, and they have seen Engel. In any historical record of influential AMVs, not including Engel would be an undesirable oversight at best.
Video: Far and Away
Creator: Maboroshi Studios
Series: Kareshi Kanojo no Jijyou
Year: 2000
Why Influential: It's long, it's instrumental, it's dramatic, and we all stared dumbfounded at it. It was one of the most amazing things I saw when I saw it at AWA 5. And the entire audience seemed to very much agree with me. You could watch it and understand everything, even if you hadn't seen the anime before. It really did a lot to show what could be done with instrumental music in a way that was dramatic and still hold the audience's attention. I also recall I hadn't seen much use of transitions up to that point by many editors. I recall mostly straight cuts, so alternating depending on the tempo of the sections really stood out. (On a side note, I used the same source, as in the same media type, as Maboroshi almost a year later to make Arima to Miyazawa).
Video: Senshi on Springer
Creator: dokidoki
Series: Sailor Moon (and Jerry Springer)
Year: 2000
Why Influential: I could say his style started with Ranma Kombat, and I could say that Right Now was even more influential. But I think what really propelled Tim to the forefront was Senshi on Springer. It made the audience already there and waiting for Right Now and set the pattern people expected from him. So I have to say, SoS is probably more influential, historically speaking.
Video: Particle Dance
Creator: Quu
Series: Shoujo Kakumei Utena
Year: 2001
Why Influential: Quu pioneered some of the ways to work with game footage, and his guide still works today for Sega Saturn footage. Particle Dance is entirely from the Utena game. It is also one of the earliest techno only videos I am aware of.
Video: I Can't Move On
Creator: ErMaC
Series: Marmalade Boy
Year: 2001
Why Influential: Although repetitive, it is sets the pattern for beat matching and lyric matching at the same time with overlays. One of the earliest examples of this. Although ErMaC doesn't like it much, it is actually my favorite video from the era.
Video: Sappy Self Indulgence
Creator: Kusoyaro (Hsien Lee)
Series: Various
Year: 2002
Why Influential: This is another one of those videos that was just talked about so much. So many people knew it. It is also one of, if not the first, example of using multiple anime and multiple characters as a whole in a way that isn't distracting but just really damn cool. Even though it jumps from character to character, it still really fits very well.
Video: Euphoria
Creator: Koopiskeva
Series: RahXephon
Year: 2003
Why Influential: KEWPTANGLES. Also, ps, perception is better. :3
Video: Orange Five-O
Creator: Vlad G Pohnert
Series: Kimagure Orange Road
Year: 2004
Why Influential: Well, aside from it being Vlad again, there are a number of things done here. Primarily it is reconstruction. Vlad takes a few key episodes of the series and turns it into a brand new opening for a TV series that doesn't exist. Due to his editing order, he creates the illusion that the anime is about something very different than it actually is. A style of editing that, for a while, definitely caught on.
Video: Waking Hour
Creator: Koopiskeva
Series: Beyond the Clouds
Year: 2005
Why Influential: Elevating the drama to a level that really hadn't been accomplished before. Lots of reliance on internal sync and less is more editing, in direct contrast to his previous works (except Perception!). Another one of those videos that just got talked about and spread quickly.
At this point, I think the rest are fairly obvious to most people, and well recorded in recent history.