In the brief time I've played 5e, I've found one particular situation in which not being able to delay my place in the initiative is incredibly annoying. I'm playing a Rogue|Arcane Trickster with an Owl familiar. The intent is for my familiar to use the Help action to grant me advantage on my attacks. With the Owl's flyby ability, it can move 30', use the action to grant advantage and fly back out 30' without drawing an OA. Pretty cool when it works, as it very reliably sets up sneak attack at just 3rd level.
However, in combats where initiative rolls split us up, it often doesn't work at all. The worst case is when the Owl rolls initiative shortly after me. Because I go first, I make my attack without advantage. The Owl follows, granting advantage on the next ally attack against the target. That can leave as many as four party members to follow up, one of whom is pretty likely to take a shot at my target, and also the enemy itself, which might simply move out of range before I come back around in the initiative sequence.
Granting advantage to a teammate is hardly the worst thing that can happen, but it's generally better for me to have it than anyone else since I get to apply extra damage on the hit. And it also kind of sucks that my familiar, which I'm in constant mental contact with, can't be better coordinated with my own actions.
RAW, the only thing I can do to guarantee I get advantage in those cases is to take a Ready action, but that's an awful option as it costs me both my Cunning Action to Disengage/Hide after an attack as well as any Reaction I might need to take on defense (Uncanny Dodge, etc.)
TL;DR: It'd be swell if I could make sure I go after my own familiar in the initiative order, but the rules really don't take that into account.