Cost to start quest: 1500 Myth (three 500 mythology inks) to obtain a Sphere Scroll of which you need only one per weapon. The scroll is used to fill out stats for your weapon. The stats are not added until the scroll is complete.

NPC: Hubairtin

Location: Central Thanalan Materia Camp

Your base stats (VIT, MND, DEX, STR, or INT) upgrade automatically. You don't need to worry about them for this. You can upgrade five additional bonus stats for 75 points total (PIE, CRT, DTM, SPS, SKP, PRY) which replace your current additional bonus stats of the animus weapon:

Main Stat Maximum: + 44

Sub Stat Maximum: +31

Accuracy, spell speed, skill speed, crit and parry cap at 44 points. Piety caps at 33 points. Determination caps at 31 points.

Example

Each upgrade is done with an alexandrite and materia. Alternate stats that you didn't choose are removed when upgrade is complete (possibly sooner). You must go through this process, even if you want to maintain the current stats when you upgrade.
Alexandrite

You need exactly 75 alexandrite total. A failed meld does not result in the loss of an alexandrite. They are acquired though the following ways:

Atma-style fate drops (must wear animus weapon).

Completing an alexandrite map (key item) obtained from using mysterious map (unique untradable item). These function the same as other treasure maps. If you fail, you lose the map. Others may help but one alexandrite is rewarded per map for the item holder alone. You can acquire a mysterious map in the following ways:
Completing the daily quest "Morbid Motivation". This quest requires you to do an expert roulette (you do not need to wear your weapon for the roulette), where you will be rewarded with a mysterious map. The quest resets in the same manner roulettes do. If you don't complete the quest by the roulette reset time, you will not be able to start a new quest that day.
NPC: Brangwine
Location: Revenant's Toll
Purchase for 800 mythology.
NPC: Auriana
Location: Revenant's Toll

Update v2.3 will add another way to obtain alexandrite.

Materia

Materia is used for each meld (75 successful melds required for upgrade). Regardless of materia stats, you will gain only one point for each meld. If the melding fails, the materia is likely lost but you will keep the alexandrite catalyst. The materia tier required rises the further you progress in a particular stat.
Melding

Melding can fail. The chance of success is given before attempt. It starts with a 100% success rate that drops as you progress higher in a particular stat. When a higher tier materia is required, the success rate returns to 100% and falls similarly to the last tier. You do not need crafting skills to meld. You will keep your alexandrite in a failed meld but likely lose your materia.