This is what is bizarre to me. That power exists in the game already, for all players. If you happen to roll where you'd prefer to go, the ability works. So, it can't be balanced around that concept - you can't assume where the player will be going and they might well be going after their ally takes a turn.
The concept of First Strike already exists in RPG lexicon, so I suppose the concept I am talking about here is Last Strike, or Later Strike. The ability to move your initiative count down one time, and only in the first round. It doesn't delay things, it doesn't cause a hideous amount of more tactical play in the game, and I really don't see how it's unbalancing given it can already work out that way so it can't be for balance reasons that the current rules are written that way.
Using the term "Delay" implies 3E or 4E Delay as written, so when you use that term, do not be surprise if people actually start talking about Delay and not "only first round Delay" or even "only when dice are first rolled declare your lower init number Delay". These are different things as well with normal Delay being more useful/potent than Delay in Round One, and Delay in Round One being more useful/potent than Lower Init Pre-Combat.
Also when talking about normal Delay (which he was talking about), there's a difference. In the one case, the player cannot just take his turn when it is most optimal to do so. He is forced to deal (or not deal) with the scenario as is. In the other case, the player can take his turn when it is most optimal to do so. The latter is not just a case of performing actions when it is most optimal to do so, but it is also a case of performing the perfect action at the perfect time and can easily interrupt enemy plans that the former cannot. If the NPCs plan is to drop a PC unconscious followed by the next NPC in init doing a coup de grace, that plan can easily be thwarted with Delay.
The former is less potent with less utility, the latter is more potent with more utility, but at a slight cost.
Some people find the latter to be unbalanced because it is not the same. It's really not that bizarre of a concept.
And quite frankly, if Delay or Delay Round One or Lower Init Pre-Combat works for your group, go for it. Just because some people find the first two of these to be a bit more implausible and potent doesn't mean that you should not use them for your game.