[Patch 4.10] New Blue Ezreal
With the new AD items, I think Blue Ezreal, or at least the idea of "Blue Ezreal," will make a resurgence, but not as an ADC. I'll explain why not as an ADC later, but I want to go through the build first.
The old Blue Ezreal that was popular mid season 3 consisted of Manamune, Spirit of the Elder Lizard, Frozen Gauntlets, cdr boots, Last Whisper, BotRK. This build worked because of not only the meta at the time, but also the items were a little different. Tear was a little easier to stack, and the Spirit Stone items gave flat mana and health regen. However, it was a really slow build, and still is due to the both the cost and the need to stack Manamune. After the Spirit Stone items were changed so that they couldn't be built in lane (only give regen by attacking jungle creeps), many people stopped playing Blue Ezreal, but others simply adapted and built Black Cleaver instead. That iteration has kind of been the standard Blue Build for awhile. However, with the new AD item changes, the Blue Build is seeing a little change.
I'm not sure what other people are building, but this is what I think to be the best option. Essence Reaver > Brutalizer > CDR Boots > Frozen Gauntlets > Black Cleaver > Last Whisper > BT/BotRK. I think Essence Reaver is a really awesome sustain item that is cheap but effective. It doesn't give the same late game power that Muramana gives, but it allows for an earlier power spike, and gives pretty much every stat that you want. The sustain is pretty solid on Essence Reaver, and so gives Blue Ezreal the ability to sustain up after long trades. Brutalizer/Black Cleaver is a necessary second then 4th purchase since you will be lacking in damage. Brutalizer gives flat armor pen, so early in the game, when your opponent doesn't have much armor, you will deal more damage. Additionally, Black Cleaver's passive is really good in team fights. Frosty Mittens give Blue Ezreal his kiting power and are the real reason why this build works. With these, you can kite for days, and are basically the hardest champ in the game to catch. With max cdr, your E is on about a 6 sec cd, and you can throw out a Q about once every 2 sec. And that is exactly why this build is popular/attractive. It gives Ezreal so much utility. Anyways, Last Whisper is a must at some point since it gives plenty of armor shred, and will let you deal with tanks a little easier. Pretty standard stuff. Finally, the defensive/last item. This really depends on what you like in a build/what the enemy team composition is. If you like greater kiting power, BotRK is the way to go. If you want stronger burst, BT is the way to go. If they have one or two big tanks, you probably want BotRK for the hp shred. If they are all glass cannons, BT will give you a passive shield to block some of the damage, but you'll have slightly stronger burst. I actually like the new BT passive, but after reading up on some of the numbers, BotRK does a lot more damage in sustained fights than BT due to the %HP damage. However, you can always get a Banshee's or GA, but I think a lifesteal item is better for the purposes of the build.
Where should you take the new Blue Ezreal?
Do not play Blue Ezreal as an ADC. Building Essence Reaver first will put you at a huge disadvantage in terms of overall damage output. The build also doesn't really spike until 3-4 items in. You can build the Frosty Mitts second for greater defense, but a standard ADC build will straight up out class you. As soon as you fall behind with Blue Ezreal, you will stay behind. That has been the downside of this build for the longest time, and building it in the ADC role is just asking to lose lane.
Do play Blue Ezreal as a solo laner, most likely mid, or possibly top depending on the match up. Both lanes tend to be farm heavy and take awhile to scale up. The Blue build also has kiting potential in it, so laning against a tanky top laner isn't the worst thing in the world, since you can kite backwards if they engage. Ezreal has decent wave clear and solid poke, so playing in a solo lane lets him safely farm up and get all the items he wants. Plus, you can usually pick up a kill here or there. His range is hard for some mid laners to deal with, and can be a weird match up if you don't usually play against him. Also, your low damage won't be a big issue since many mid laners start with some form of mana sustain, and don't usually get to much damage until Deathcap, so you have some time to scale up. Also, ask your jungler for ganks to help get you going. A few early kills can make this slower build go much faster, and be much more effective.
New Blue Ezreal Build: ER > FG > CDR Boots> BC > LW > BotRK. Play him in a solo lane, not as an ADC.