|How to Start:||Talk to Ellie after traveling to Byzantium|
After traveling to Byzantium for the first time - as soon as you dock at one of Byzantium's two landing pads, in fact - Ellie will wait for you at the door and ask to talk. Hear her out and she'll ask to meet her parents, who she, by her own admission, hasn't visited in a long time. She wants you to come as well, and bring you worst foot forward. Yeah... looks like Ellie wants to involve you in some petty, juvenile dispute with her folks. Cool! You'll travel to Byzantium for the first time either after completing Balance Due or Radio Free Monarch, so the time table for this can vary significantly.
Whenever you start this quest, make your way to Byzantium's Estate District, and from the stairs leading to the Acropolis District turn north to find the Fenhill Estate. Once inside, pass through some doors to the east to encounter Ellie's - or rather, snicker, Marilyn's - parents. She'll take petty delight in showing off how uncouth she and her companions are... assuming you play the part of the boor as requested. Whether you do or not (it's almost more fun subverting Ellie's expectations), things will take a turn for the worse when Ellie realizes her parents... well, they moved on. Worse still, they profited from Ellie's prolonged absence, and have no intention of acknowledging Ellie's existence - they've got social ladders to climb, neighbors to impress!
The emotional turmoil of having parents who clearly don't care about your well-being is bad enough (not that Ellie can really complain, given her own attitude towards them...), but one has to imagine part of the reason Ellie's so upset by the turn of events is the simple fact that since she left to pursue a life of adventure as a pirate her parents have made more money from perfidy than Ellie has! It's a personal and professional insult!
When the encounter ends, leave the house and talk to Ellie, who will share her fantasy version of how things were supposed to play out (you can pass an [Intimidate 40] check here, for what it's worth). After rejecting the opportunity to learn something useful from this episode (she really is her parent's daughter...) she'll scheme about one last act of revenge. Ellie's parents clearly don't care about Ellie at all, so she decides to hit them where it'll really hurt - their wallets. Make your way to the eastern edge of Byzantium (up the stairs to the northeast of Prosperity Plaza) and search a mailbox on the side of a building down an alley to find that the Greater Halcyon Insurance Group has relocated to Fallbrook on Monarch.
Since this quest requires you to travel to Monarch, it's time for the typical "how to get to Monarch" spiel. If you're working for Phineas, of course, you'll only reach Byzantium after Monarch, but if you are working with the Board you'll either need to travel to the Cascadia Landing Pad and make the perilous run over the surface of Monarch to reach Fallbrook, which can be found along the southeastern edge of Monarch. Less risky, however, is simply completing the quest Long Distance, after which you'll acquire a Stellar Bay Navkey which will allow you to land at Stellar Bay on Monarch, constituting a much less perilous journey south to reach Fallbrook.
Once you reach Fallbrook, follow the road until you find the unmarked Greater Halcyon Insurance Group building to the north of the road - it's just across the street from the SubLight Goods & Sundry building. Talk to the Insurance Agent and pick the option "I need to change the beneficiary on Marilyn Fenhill's life insurance policy.", after which just question her until she... well, almost seems to encourage you to sneak into back room, hack a terminal, and commit a bit of profitable fraud.
If you don't want to bother with the extra leg-work, just bribe her (this set us back around 1,500 Bits) or pass an [Intimidate 30] check to bully the insurance agent into changing the policy herself. On the other hand if you're not averse to getting your hands dirty head upstairs and pick a locked door [Lockpick 45], or failing that steal the steal the Insurance Office Key from the insurance agent to bypass the Lockpick check, or exit the building and search its northwestern wall to find a ladder you can climb. Either way you manage it you should be able to access the terminal [Hack 21] upstairs, where you'll need to pick the options [Policy Details: Fenhill, Marilyn] and [Change Beneficiary To: Ellie Fenhill]. You can also read the terminal for some amusing claims. Well, maybe amusing is the wrong word.
After changing the insurance policy - however you managed it - talk to Ellie again to finish the quest. For your "work" you'll get 1,125 Bits, and a Framed Insurance Check and Ellie will earn the Beyond the Grave perk, which reduces the cost of bribes by 50%.