First off, Lux's ultimate has a serious issue that Riot seems incapable of fixing. It is *still* missing people, I had it happen twice in both games where I was playing Lux, the beam goes right over people and does no damage at all. Just keep that in mind when playing her.
Lux needs CDR. Get max CDR and her ultimate is less than 30s for CD, I believe.
So Lucid boots/Frozen Heart is good.
Always look at the enemy team, though, if they are heavy on CC, you might want Merc Treads rather than Lucid.
If you go Lucid first and then some AP, you're generally going to die if an ADC so much as sneezes on you. That's why Frozen Heart is such a core part of most Lux's I see, (and mine as well).
Rylais is something that I would say not to build on Lux, although I have seen it a few times.
The items I most see on her are:
Lucid/Sorc Boots, (Lucid being better than the +20 mrpen, IMO), Frozen Heart, Abyssal Scepter, Rabbadon's Deathcap, Athene's Unholy Grail.
Items are always situational to enemy team, but Lux needs to go for CDR and then damage, in my experience. The reason being is early game she can rely on base damages, as well as the fact that she's a great supporter early on with her shield, slow, and snare, and doesn't need to be pumping out major damage, and this allows her to keep throwing those spells.
Late game she needs the damage as well, thus adding lots of AP after your FH is completed.
Throw in an Odyn's Veil somewhere if an AP is bothering you and there's a pretty solid Lux build.