Anyway, here are seven episodes of Buffy which I'd have given Amy Madison a cameo in.
The goal isn't to change the plot of any of these episodes substantially or to change Amy's overall arc (such as it is). Rather the idea is to do just enough work to make Amy feel like a recurring character with the weight of, say, pre-S6 Jonathan Levinson or pre-AtS Harmony Kendall. Somebody who is stil clearly a minor character, but who feels a lot more thought out than Amy herself often does.
I think you can do that with just a few strategically chosen cameos. Here are my picks:
Prophecy Girl brings back Jenny Calendar (previously only seen in I Robot, You Jane) and the previous week Out of Sight, Out of Mind had brought back Harmony (previously only seen in The Harvest). Both Harmony and Jenny would go on to be popular recurring characters, and I think making a second appearance in the first season would do wonders for Amy as well.
While I'm at it, I'd slightly rework Cordelia's story this episode -- cutting out her doomed boyfriend Kevin, who replaces Out of Sight, Out of Mind's Mitch without explanation and will never be mentioned again; and the slightly uncharacteristic friendly way in which Willow and Cordelia treat each other this episode. In my version of the episode, it would be Amy who roped Willow in to help set up the sound system at the Bronze for the Spring Fling, and it would be some of Amy's friends that they found murdered by vampires in the Audio-Visual Club.
(I wouldn't rob Cordelia of any of her other scenes, of course: she'd still get to drive her car through the school halls and bite a vampire.)
I think adding Amy like this would serve to remind the viewer that (i) Amy exists; (ii) Amy and Willow are friends; (iii) even with her mother gone, horrible things keep happening to Amy Madison.
2) S2E01 When She Was Bad
Nothing fancy here; just a quick establishing scene at some point to remind us Amy still exists (in the same way Larry makes a cameo at the beginning of Season 3). Have her show up at the library and ask if any of the gang have seen Jenny Calendar, because she's meant to be teaching a class but nobody's seen her all day, maybe? Rather than the approach that canon takes, where nobody seems to realize Jenny has been kidnapped at all, even though we know she was captured at least a day before Buffy goes to rescue Cordelia (and the gang spend all of that day at school, where Jenny Calendar should be working...).
Again, the point of this addition is to mainly remind the audience that Amy exists, but we're also setting up the idea that Amy might eventually look to learn more than computer science from Ms. Calendar.
Just one scene here as well: Amy commiserating with Buffy and the others, and admitting that Snyder "volunteered" her to go out accompanying kids that evening as well. The point being that we see that Amy has dressed up as a witch for Halloween...
(We wouldn't see Amy again all episode, and we wouldn't know whether she got her costume from Ethan's, but down the line the idea that Amy might have gotten a bit of a head start as a witch from her experiences this episode -- in the same way Xander remembered some of his military knowledge -- could definitely be a fun fan theory.)
If anybody asks Amy if she's comfortable dressing up as a witch after what her mother did, maybe she can reference back to the attack on the AV club or Ms. Calendar being kidnapped and suggest that she knows now that there are worse things in the world than witches...
4) S3E02 Dead Man's Party
OK, if I were rewriting this episode I'd find it hard not to make bigger changes. At the very least, I'd want to have a few more scenes from the POV of characters other than Buffy (so that there's less of a sense of Buffy's friends all treating her horribly for no reason). And who better for Willow to be pouring her frustrations out to than her fellow fledgling witch Amy?
Give Amy a line about "missing Ms. Calendar too" as well, and Willow's final speech to Buffy about spending the summer "trying to communicate with the spirit world" take on an interesting subtext.
This is an interesting one because I'm half-convinced Amy was meant to be in this epsiode anyway. It's the only episode before Amy gets ratted in which she's mentioned on screen (Willow tells us that she's heard some gossip about Cordelia from Amy, which is -- in canon, anyway -- the first indication we get in the whole show that Amy and Willow have started hanging out).
I wouldn't be at all surprised if Nancy's role this episode (in the alternate Wish dimension) wasn't originally meant to go to Amy. That's how I'd play it, anyway. (Of course, if Buffy didn't come to Sunnydale she wouldn't have been around to save Amy from her mom. But I think that's easy enough to fix: just throw in a line about Amy being grateful to Giles and the other Whte Hats for "saving her" from some unspecific fate, and make sure to establish that this version of Amy can't do magic.)
Of course, this Amy would have the same fate as Nancy -- fittingly, killed by Willow -- but I don't think we'd need to have that happen off-screen the way it does in the actual episode.
I really don't think Amy's abrupt descent into magical drug addict works ... well, at all. So I'd like to see a much bigger change to her arc after being de-ratted (something like @idle-flower suggests here, for example). I just don't think her villain arc this season makes any sense (and I know it robs Willow of a lot of the moral agency she should have at this point).
Failing that sort of change, though, I'd at least like Amy's mini-arc this season to .. resolve? She gets the door slammed on her face by Willow all the way back in Doublemeat Palace and then we never see her again. We don't even know if she's staying with her dad, or if her dad left town, or what. How is she surviving at all? At least throw in a scene of Amy discovering whatever's left of Rack and looking angry and vengeful about it, surely? Remind us she exists, and that things aren't going great for her.
7) S7E12 The Killer In Me
You might object to this one on the grounds that Amy is already in this episode.
The thing is though, I'm not sure that episode's writers remember that she is? Amy is outed as being responsible for Willow's predicament, Kennedy promises to stop her, Amy snaps her fingers -- teleporting Kennedy away -- and ... that's it. That's the last we hear from, or about, Amy Madison. Not just in the episode, but in the whole show. She doesn't suffer any consequences, she never gets a face to face confrontation with Willow, there's no obvious reason why she wouldn't be free to try again and again. Surely we can do better than that?
At least give us a scene of Amy fleeing Sunnydale ahead of the First's wave of destruction, or reacting to Willow getting her own body back. Something that we can pretend amounts to closure for a character who has, after all, been a part of the show since the third ever episode. Even if her continued appearance on the show, season after season, wasn't ever something the writers conciously planned.