"Abolish" is a term from FF8, when the Queen of Cards travels around to different cities. You can petition her to add or abolish a rule in the current region that she is in. So "Abolish Random" is what you chose when you want her to get rid of that rule. Just google the phrase and you'll see hundreds of forums threads and FAQs concerning this mechanic in FF8. The OP was just referring to that, and the seemingly overwhelming consensus on the internet that the random rule in FF8 was horrible. Now calm down.
There were many ways to deal with the random rule in FF8. You could delete or convert all of your non-unique cards, meaning that the possible pool of cards in your hand were all powerful (what would count as 5 star in this game). Opponents had card decks with a dozen or more cards, many of them trash, so that random rule would affect NPCs just as much as players. And of course, you could SAVE YOUR GAME after every win, and reload after every loss. That was because you would LOSE CARDS every time you lost, and the NPC opponent would always select your irreplaceable cards and then use them against you in subsequent matches.
In FFXIV, we have no way of dealing with random other than to suffer repeated bad luck. NPCs aren't affected by random either, as Landenel in South Shroud hits me with an Odin + Bahumat combo on this last two turns nearly every time I play him, which isn't even possible for a player built deck. What makes this bearable is that we don't lose cards. Just imagine Landenel stealing all of your Platinum cards and turning them against you... you would never win again.
Now... a possible idea is to have a card game npc using roulette rules in each region. Beating her in a duel would allow you to change the regional rule at random, but only affecting yourself. Essentially it would be like the "abolish random" feature in FF8, with some RNG added to the mix. It would liven things up, make seemingly impossible duels less harsh on certain deck types, and allow you to play some of the rare rule sets if you so choose.