Concrete suggestions, assuming you don't want core/RAW:
1) Super simple: No rolling, just go in order of Dexterity. In case of a tie, then you roll a D20, no modifiers. Roll until no ties. Just write down the names on a sheet of paper. It stays the same the whole combat.
2) Mechanically simpler, potentially hazardous: Go by side (PCs and allies vs. "monsters") -- use highest dexterity per side. Each side gets to internally negotiate exactly how they want to order. I'd avoid this, though, because all your giants will be hiring goblin retainers, just to get the initiative.
3) Simple randomization: Use a deck of cards. Deal one per unique combatant or group of mooks. Leave the cards, face up, in front of each "owner". Step from high to low, Joker is wild. Ignore modifiers, since it ruins the display (cost of simplicity).
3a) Cards with modifiers: If ignoring the modifiers bothers you, then deal each player a hand of cards at the beginning of the session. The hand size is equal to the absolute value of their Dexterity modifier. After drawing for initiative at the top of combat, each player is permitted (dex bonus) or required (dex penalty) to use a card of their choice (bonus) or the lowest card (penalty) from their hand, in place of the drawn card. In this case, the drawn card goes into their hand. In this way, they always have the same number of cards in their hand. If you have marathon sessions, collect the hands, shuffle, and re-deal after roughly four hours of play.
4) Complex and "realistic": If you can get your hands on a copy of Phoenix Command and the melee combat supplement, go with that. Since I haven't seen them since the 1980s, you're probably going to have to roll your own. Every action has a speed factor. You can use 1E AD&D speed factors, if available, or you can do something more simplistic. For example, weapons have a 4 base SF. Weapons with the "two-handed" (or versatile, used two-handed) or "heavy" properties get a +2 SF for each. The "light" and "finesse" properties each shift the SF by -1. Yes, that difference is intentional. Spells that take an action to cast are base of 2 SF, +1 per spell level (yes,
meteor swarm has the potential to set you up for real pain). Spells that can be cast as a bonus action are automatically a 2 SF or 4 SF, if 5th level or above (balance-ish). Movement is 1 SF per 5' moved, with faster and slower creatures having a slightly different rate (I have ideas, but they're complex). Combat is assumed to start on segment 10, with the initiative modifier being subtracted from that for when a given character starts. Armor limits dexterity for this purpose. On their number, each combatant chooses an action and begins to execute it. Their action completes in a number of segments equal to the SF of the action. At that point, they choose a new action. Readied actions begin one segment after the triggering action begins. If you're just delaying (resurrecting that option), it costs you 5 segments to jump back into the fray.
4a) Complex with better modifiers: If you like 4, but want Dexterity to matter more, that can be worked in pretty easily. Just add 4 to the SF of all actions. Now, subtract the dex modifier from the SF every time. An action can never take less than 1 segment. You could get wild with this and use Intelligence to modify the SF for spells, Strength for weapons, etc. Personally, I'd avoid this other than to limit the modifier to the
lesser of Dexterity or attack stat. You could either leave armor to limit dexterity impact on SF or you could apply a +1 SF for medium armor and a +2 SF for heavy armor, with non-proficiency doubling it. Each level of encumbrance is +2 to all SF.
As for my thoughts on what to use, I'd still go with the PHB rule. I don't see #1 being much quicker to collect each value vs. including a roll. I use a combat tracker that can roll for me, so that's totally no gain. I do
not like #2, at all. It was included because some folks like initiative by side. I actually kinda like the flavor of using cards, but I thing it's equal in complexity to the PHB rules, so it's just aesthetics. If I did use cards, I'd probably go with #3a, just for fun. Playing Phoenix Command cured me of any complaints I had around "realism". I know I would not enjoy #4, but I also know it's totally workable as long as you like math.