Lux has always been one of my favorite champions. Her spells are some of the most powerful in the game in my opinion. She can snare. She can slow. She can nuke. She can snipe. Last and also least she can shield.
A lot of people make the big mistake of trying to play Lux defensively. Don't waste your gold on catalyst or any other health items early / mid game, Lux is an offensive champion. The best defense is a good offense with her more than any other champion. You don't need to worry about having health items if none of your opponents ever get to you.

I use clarity because Lux is very mana hungry early game, and I miss out on kills and precious stacks when I have to go back just for mana.
Flash is self explanatory.
I wouldn't trade flash for any other summoner skill ever, but I sometimes swap clarity for the following:

• Ignite - Good for taking down carry's like Master Yi that have a lot of lifesteal/spellvamp or other healing abilities. Also good for picking up kills early game.
• Teleport - Early game: Teleport is just like clarity except it takes a little more time. Because it gives you the ability to go back, heal and regenerate mana, buy items, then return to your lane before a single minion has died. Mid/Late game: Teleport gives you map control, and the ability to get into position for Snipes or team fights without having to run across the entire map to do so.
• Exhaust - Personally I don't think exhaust really fits Lux's playstyle quite as much as any of the above summoner spells. Though it isn't bad if you're new to Lux, and want to be able to win most 1v1's. I don't need Exhaust to win 1v1's, if I used it, it would be to nuder an AD carry during teamfights in order to take him down. I guess you could get improved Exhaust in masteries, then use it to ensure kills by dropping it on someone who is within ult range in order to prevent them from escaping the line of fire, while simultaneously reducing their magic resistance... seems a little much for Lux though, just keep her tank full and the kill come easily.
• Ghost - I don't see the need for ghost and flash, if you prefer ghost to flash for some reason then feel free to interchange. I feel that they fill the same role: Escape / Strategic Positioning / Lunge for kill. I also don't think they would be bad together, ghost / flash is quite often the norm for most champions in more competitive play. For Lux, mobility and positioning are key.
• anything else - Good luck.

I took the philly stone rush from another build I admit it, I hadn't tried it but once I did I couldn't build her any other way.
However I have always rushed a Mejai's Soulstealer and Rabadon's Deathcap (used to be Zhonya's ring) with Sorcerer's Shoes and a Lich Bane. Lux is a grade a nuker. Lich bane combined with Lux's passive is just godly late game, depending on how many stacks you have. Usually I get lich bane after Deathcap because I have enough stacks for it to be worth it, if not then I pick up Void Staff and Rylai's Crystal Scepter for added AP/survival.
The Elixir of Brilliance right after Mejai's Soulstealer is to make up for the Soulstealer being weak when I first get it. How this works is after I get Mejai's I chug a potion to jumpstart my soul-stealing.
Basically when it comes to item stats, Ability Power should be your only concern.
Item Stat Priority
Early Game Item Stat Priority:

1. Health Regen
2. Mana regen
3. Move speed
4. Ability Power

Mid Game Item Stat Priority

1. Ability Power
2. Ability Power
3. Ability Power
4. Ability Power

Late Game Item Stat Priority

1. Spell Penetration
2. Health
3. Lich Bane Passive
4. Rylai's Crystal Scepter Passive
5. Zhonya's Hourglass Active(if needed)
6. Banshee's Veil passive(if needed)

I myself have never gotten banshees veil on Lux before, although I can think of many situations where I probably should have.
As for Zhonya's Hourglass; I have used it on a few occasions, usually in games where an AP carry (Such as: Katarina, Nunu, Karthus, Malzahar, Amumu or even another Lux) is steamrolling me with an ult that the activated ability on Zhonya's Hourglass could save me from. This is especially fun against Karthus, because he thinks he's got another easy kill in the bag until you Zhonya's Hourglass perfectly through his ult. Remember though, Timing is everything with this item. I have never picked it up for the armor though considering the hefty amount of ability power on it,(100) I may consider buying it more often against AD carry's.

Pros:

• Extremely Long Range
• Boss Damage
• Crowd Control / AOE / Multi-Target Nuke's
• Low Ult CoolDown
• She's Cute

Cons:

• Extremely Squishy
• Very Unforgiving Spells(They're all skill shots with relatively long cooldowns)
• If You Miss Light Binding On Xin Zhao(any melee carry), He Will Kill You.
• Unique Playstyle - It takes some getting used to.

She can influence team fights immensely without even being close. All of her spells are designed to keep the enemy at bay while inflicting critical damage. During early game (with some harassment) by the time anyone gets within melee range of you, your passive and an auto attack is all it takes to kill them

Lux is probably the worst jungler in the history of leagues. Though if you can manage to pick up a golem buff yourself or have a teammate help take it down its a huge advantage for Lux. Possibly more powerful on Lux than any other champion IMO. Once you get max cooldown reduction you can let an ult loose every 24 seconds. So the cooldown reduction combined with the mana regen is everything a good Lux will need from a buff.