They are often used as plot hooks themselves - sometimes more than they should be, as Mentors and Allies and Retainers the players have paid for should be helpful more often than they need helping, or it gets a bit like “stop hitting yourself” bullying from the Storyteller. This is where loner PCs with no friends or family come from. Backgrounds are designed to key the PCs into the world around them, so should not be abused...
A good friend will help you move, but a true friend will help you move a body.
Steven J. Daniels, Weeds In The Garden Of Love
Equally, at a high enough level they’re open to abuse by the players. A PC with Resources 5 can buy her way out of a lot of problems, and a PC with Allies 5 could swap his character out for someone more useful for a given scene quite often.
But this can go a long way to defining a character, more than most trait sets. A player putting a lot into Resources probably wants their character to deal with high society, big business or the upper echelons of crime, and probably look good doing it. Fame can be a double-edged sword for a vampire, but can provide a different view of the setting than the anonymity PCs normally have.
So do I have a favourite? Well, I usually take some Resources so I’m not playing a crazy homeless vampire... I’d say Contacts, the handy catch-all adds-ways-of-looking-into-things Background. or one of the pseudo-Background-y Merits like having your own nightclub. Because being able to own the bar that all the PCs hang out in is just kinda cool.
I do have another favourite in the wider Storyteller systems, though - Nemesis in Adventure! is essentially the classic Enemy “disadvantage” where you get to define a recurring foe that the Storyteller is then expected to use, turned into something to buy as it gives you bonuses when fighting them. That’s brilliant.