A hearty, meaty lasagna is one of my go-to meals when I’m having company over. I precede it with a green salad, and follow it with coffee and dessert. As a former Southerner, of course I also have bread-and-butter pickles on the table. But the squares of lasagna look rather lonely on peoples’ plates.
I used to make sure I had garlic toast, but that seems like redundant carbs with lasagna. So what would a good Italian mother serve with her lasagna?
A nice Italian salad. Just oil, balsamic, and seasonings. Lettuce, tomatoes, etc.
Light and healthful.
I am a sucker for good lasagna.
Yeah, I would want the salad with the lasagna.
Salad and garlic bread.
Chianti
That is like Milo’s car. 
Sounds like those plates need another piece of lasagna. 
But yes, something bright and acidic to cut through the fat and meaty richness. Salad wirh a vinegar-based dressing is a good call.
Something like asparagus roasted with lemons could be nice, or Brussels sprouts in a balsamic reduction.
A stout or a barley wine.
A light salad, light vinaigrette is perfect. Of course, it better have bread involved somehow. Not croutons. That’s not bread. Keep up with th garlic bread. 
Yes, lasagna needs something, and that something is invariably a salad. My go-to has always been well-anointed garlicky Caesar salad. I serve both together on large plates. The nourishment, taste, and appearance of lasagna alongside a strongly flavoured robust Caesar salad are a perfect combination in every respect! And of course lots of red wine, usually Cabernet.
Garlic bread is a nice touch, too, but optional.
I’m making lasagna for our Easter dinner. One sister is bringing a big green salad and another sister is bringing garlic bread. I’ll also have some fruit on the table. That’s why I like lasagna - you don’t have to have numerous side dishes. It’s simple but so good. Of course, there will be a few desserts also.
Since the question seems as much about plating as it does about flavor/contrasts, I’d suggest julienned roasted orange and yellow bell peppers (red are my normal choice, but want the color contrast here) marinated in a good cider vinegar (a good balsamic would be a normal choice, but I’d want something slightly less sweet with a marinara lasagna).
The two other main options I would suggest is an artichoke dish - you can go super easy with quality marinated artichoke heart quarters from the jar, or make the equivalent yourself. The astringency is a bit much for some people though, and may overemphasize any tomato based sweetness in the lasagna if you’re alternating.
So the least offensive but still great option would be a roasted or marinated mushroom dish, either small whole or sliced. The variety will depend upon tastes, but with a hearty meat based lasagna, I’d probably go with a milder option like traditional white mushrooms. Again, a simple vinegar and/or butter dressing and a chiffonade of chives or other green herb should give a great contrast of flavor, color and texture. And if served cold, can be a temperature contrast as well.
@Johnny_Bravo’s suggestion of asparagus or brussels would be another excellent choice.
What to serve? More lasagna.
Roasted brussels sprouts are a good idea too.
For God’s sake, keep that fat-ass, orange tabby cat off the table!
Salad, marinated veg and definitely garlic bread, knots, or sticks.
Usually a vegetable, garlic bread, and a salad.
So - prosecco.
It’s always the right time for bubbles.
I was going to say you need smaller plates, but this is a better answer.
Coincidental reddit thread with the very same question today. I like the cucumber salad and bruschetta suggestions.