Get a shot off fast. This upsets him long enough to let you make your second shot perfect.- Robert Heinlein
As a simple principle, I want the players to be able to make a choice between accuracy, speed, and perhaps power (damage). It may be better to get a wild attack in first, or maybe you need to take your time to line up a fight-ending hit on a tough foe.
Here's a simple rule that would grab some of that idea:
"If not surprised, on the first round of combat, gain +1 to an initiative check (D6), +2 to hit (D20), or +1d6 per three levels damage on any one attack."
This is the carrot approach in that it makes PCs more powerful, unless, of course, you give such an ability to the monsters as well. You could also take another approach, allowing the PCs to "wager" for initiative:
"If not surprised, on the first round of initiative, each side may accept a penalty to hit or damage for the first round of combat. For every -2 to hit or -1d6 per three levels damage penalty, each side gains +1 to an initiative check."
However, subtraction is always harder than addition, so I'd be wary of this approach. One could also invent some sort of "penalty token" that could be gained in various ways (attacks by enemies, curses, fatigue, etc); maybe the penalty tokens give -2 to hit each. In that case you could say, "Gain one penalty token but gain bonus to initiative."
Let's assume we go with the first option -- it seems simplest, and will gain little resistance as it gives the PCs a nice little benefit. Clearly I would not want to reroll initiative every round with this. However, I do want some option to mix things up in the midst of a fight. I see two ways to do this:
(A) Every few rounds, recheck initiative. The easiest way to do this is probably to roll 1d6 at the end of each round and if it comes up as a 1 (or 1-2), it is time to reroll initiative. You could also tie this to spell duration and stop having to track spells in rounds, if you like.(B) PCs and major NPCs can force new initiative checks. For example, maybe a player can spend a full round action to force a new initiative check for all players. Or, if characters have some sort of "encounter power" (don't get all 4E on me -- remember how a PC could charge only once every turn in 1E? That's arguably an "encounter power," right?) that can be expended on a variety of things, including a new initiative check.
I think either would work fairly well--of course, this needs to be playtested.