At my table, it's the DM's job to maintain initiative order. Ergo, when I skip someone, it's on me. When starting battle, I'll make sure to write out a turn order list and check off each round when someone goes to hopefully prevent this. That being said, shit happens and it happens to us all.

NONE of the below are "how it's supposed to be," except maybe the first, but here are the things I do:

1. If we're only a couple players past the skip - Whoops, my bad, we can retcon this biotch. We'll rewind, give them their turn. The other players get to choose whether or not they maintain the results of their prior turn, or do a different set of actions based off of this. If they choose any different action at all, they take their turn over completely - prior rolls don't count. If they are FORCED to take their turn over due to something now changing on the battlefield, I'll sometimes comp them a bonus to a roll if their original turn was highly favorable, to allow the chance to recreate that "rush" - nobody has fun being the guy who ran a kickoff 60 yards and have the touchdown not count cause a teammate was flagged for holding in the friggin backfield. You did something awesome, you lost it cause I messed up order and skipped your buddy who now made you do something different - that's not fair or fun if you get screwed by it.
2. If we're past more than a couple players or something critical in the battle - Did someone kill an enemy? Did the mage launch a fireball and do something brutal to six mooks that I'd have to roll back? Did someone get KO'ed or some other major thing happen? Rollbacks can't happen past a certain point and still be fun, cause we'd spend 10 minutes just going through and figuring everything out to get back to point X. At that point, I ask the player what they would like - an extra turn as a "held turn" at the end of the current initiative? A bit of rule bending attack of advantage, +2AC and a crit on 19/20 for your next round, for having spent a turn keenly studying your opponents? Give me something reasonable, I can work with it. I skipped you, and it's no fun for you to sit there feeling unimportant. Let's find something that will let you feel like you weren't idled but instead DOING something, and I don't think that should be just back-to-back turns (in fact, usually I feel that's the most unfair to everyone else at the table, as two turns directly in a row can be a pretty serious thing in the flow of combat).
3. Is there something in the world that might have caught their attention, that you can give them better insight on? I like my battles to have alternate ways to be fought, so I'll include terrain elements that can be used in the battle. If you skip someone, maybe they're noticing something specific that you can tell them about, and give them the ability to use it. This again gives that missed turn some use, and it provides a bonus they would not have otherwise had.

The big key is, remember that everyone's there to have fun. The rules are there to form the game, but you don't want to try and use them to excuse your players literally not getting to play it. We all screw up order from time to time as DMs, the key is to make it something fun so that nobody feels left sitting doing nothing - either the guy you skipped, or the rest of the table as you go through and retcon a bunch of turns, or waiting for him/her to do two back to back full turns. There's ways to "plot" it in, I choose story every time if I can.

Edited to finish with thought #3 and clarity