What are the tasteful herbs served with dizi in Iran?

I presume it was mint. There are a lot of different variations of mint, from really spicy peppermint down to fruity flavors such as strawberry.

You probably had the Moroccan mint / Mentha spicata var. crispa. If you have specialty stores for Mideastern cuisine ask for "Nane"

Edit, after your addition of the Picture: By the looks I would suggest Thai basil, although this does not really have a minty taste and the stems should be darker ... or basil mint

Hard to say as both plants have so many variations to them. If it hat a distinct minty tase, I´d look into one of the mint variations.

And additional edit: Just hat a look on my own herbs, and realized that the young Thai Basil I planted looks exactly like your picture, light greeen stems and all:

Iranian basil. It's got a taste that's distinct from italian or thai basil. I've never gotten basil that tastes quite the same in a US grocery store, but there are a few varieties that taste similar. Apparently you can buy seeds online and grow your own though.

Sabzi khordan will also often include mint, tarragon, parsley, chives, spring onions, or radishes as well.