There is no way to unlock Mythic Weapons (yet) on ERA.  However we made it possible to unlock the Weapon Skills in a custom event.

To start the event go see "Edlina" in Upper Jeuno.  She will give you some information.  You will have to trade her 500k to obtain the key item "Dawn phantom gem".  

In the "Ra'Kaznar Inner Court" There will be Three "???" Clicking one with the "Dawn phantom gem" in possession will spawn a HYPER NOTORIOUS B I G WHAT WHAT Back from the dead at it again baaabbbyy. Anyway...You will really spawn one of the three HNM's so make sure your group is Prepared to fight to the death!

All three of the HNM's has a chance to drop every Mythic Weapon. I believe the cap on max drop is around five, and there is a chance to get zero. With TH4, you will see two+ weapons drop most of the time. Some weapons drop more frequently depending on the HNM your killing.

HNM POS Bonus Drop Rate
Whitenoise Bats J-8 Main Island THF NIN RNG PUP COR SMN
Pox Hound and Fluterfly M-9 Second Island WAR SAM DNC RDM WHM SCH BST
Wayward Bhoot and Dolorous Cyhiraeth L-10 Second Island PLD BLM DRK BRD DRG BLU

Once you obtain a Mythic Weapon, you can start unlocking the Mythical Weapon Skill. To do this you have to equip the Weapon and spam Weapon Skills to earn Points. The mob has to be Lvl 76+.  You also have to use certain Weapon Skills to gain Points. 

You have to earn 500 Weapon Skill Points. You will get a system message with your count. Not every weapon skill works towards the points.

List of Weapon Skills that counts towards Points + Info.

Job Mythic Weapon Weapon Skills Used For Gaining Points Mythic Weapon Skill Reward
Warrior Conqueror Raging Rush & Upheavel King's Justice
Monk Glanzfaust Assuran Fists & Victory Smite Ascetic's Fury
White Mage Yagrush Hexa Strike & Realrazer Mystic Boon
Black Mage Laevateinn Spirit Taker & Myrkr Vidohunir
Red Mage Murgleis Vorpal Blade & Requiescat Death Blossom
Thief Vajra Evisceration & Exenterator Mandalic Stab
Paladin Burtgang Vorpal Blade & Requiescat Atonement
Dark Knight Liberator Guillotine & Entropy Insurgency
Beastmaster Aymur Rampage & Cloudsplitter Primal Rend
Bard Carnwenhan Evisceration & Exenterator Mordant Rime
Ranger Gastraphetes Slug Shot & Last Stand Trueflight
Samurai Kogarasumaru Tachi:Kasha & Tachi:Shoha Tachi: Rana
Ninja Nagi Blade:Jin & Blade:Shun Blade: Kamu
Dragoon Ryunohigel Penta Thrust & Star Diver Drakesbane
Summoner Nirvana Teen Spirit Spirit Taker & Myrkr Garland of Bliss
Blue Mage Tizona Vorpal Blade & Requiescat Expiacion
Corsair Death Penalty Slug Shot & Last Stand Leaden Salute
Puppetmaster Kenkonken Assuran Fists & Victory Smite Stringing Pummel
Dancer Terpsichore Evisceration & Exenterator Pyrrhic Kleos
Scholar Tupsimati Spirit Taker & Myrkr Omniscience

Most players farm points at the Greater Colibri camp. This way you can obtain Zeni, Merits, and unlock your Weapon Skill all at the same time. Another popular area are elementals in SKY.