Start items vary on lane match up Either boots + 3 red pot or Flask + 3 pot
Depending on lane you either go double doran ring into DC or Athene's into DC
Use Sorc boots, and get Void Staff
The only four items you will be shooting for are Sorc boots, Death Cap, Void Staff, Athene's Unholy Grail From there it is situational
Hope this helps^^
No one builds Athene's anymore, it's garbage.
Athene's is gotten in 45% of Lux games and gives a 59% win rate. Morellonomicon isn't even in her top 4 most used items. http://www.lolking.net/champions/lux
The reason for this is because Athene's offers everything you need from CDR to fixing all mana issues. Running ignite means that the Grievous wounds on Morellonomicon is a non-issue. I don't think Morellonomicon is a bad item since definitely fits most of the criteria for what she needs. But overall Athene's is a better choice. In earlier game settings you will want to get in auto attack procs for your passive for extra harass which the MR from Athene's aids with.
Tear fills too slowly on Lux and Athene's solves all the mana issues. Additionally Lich bane brings you too close to the fight during end game and items that give higher AP would be more suitable to her role. You can definitely play her with Morellonomicon and Lich Bane but I wouldn't suggest it.
Anyone mentioning tear is bad, anyone suggesting rushing Athene's is bad, Anyone mentioning Lich Bane is bad.
If you got mana problems just get the little grail and then go straight rabadon's.
Order of core items is situational and sometimes comes down to playstyle. Unholy Grail gives CDR and permanently fixes her mana issues after a burst kill allowing her to follow up for sequential kills. If you play with a lockdown Lux playstyle Unholy Grail is not a bad first item. If you play an assassination Lux DC first is not a bad item. As suggested I mentioned that there are different builds depending on the situation although the core four items are the same.