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: