Iconic? Does it matter which country? Because there's both the Korean and Thai Full House--Full House Thai was popular in its time, but I think it's more iconic among international fans. I know it's the lakorn that introduced me to Thai dramas, as is the case with a lot of foreign viewers. I didn't even know lakorns existed until I ran across it on youtube
The Korean/original version will always be iconic, as it contributed to the spread of Hallyu. Goong/Princess Hours (NOT the Thai version-- that one sucked) would definitely be considered iconic. My Princess, Coffee Prince, Secret Garden, Sungkyunkwan Scandal (all Korean) are iconic and AWESOME dramas. I know Descendants of the Sun was insanely popular, but Idk if it'll be considered iconic.
Taiwan's It Started With A Kiss and the sequel They Kiss Again are definitely iconic. The story itself, Itazura Na Kiss, was a manga series that was made into an anime, three Japanese dramas, three Taiwanese dramas, a Korean drama, and a Thai drama. I thought the original Taiwanese version was the best, but others like the Thai version.
As for iconic Thai dramas, I don't know if I'm fully qualified to discuss them, since I usually shun the older dramas. The new ones are more westernized, which makes them more understandable and relatable for me. I speculate that the 4 Huajai Haeng Koon Kao series could be considered iconic, and perhaps Anne's Sood Sanaeha. Kluen Cheewit 2017 is iconic among international fans, I know it was a number of foreigners' first Thai drama. Perhaps Pin Anong as well, and Tawipob.
Now that you mention it, I'm surprised that Thailand hasn't remade Boys Over Flowers. They've done Coffee Prince, I know, Itazura Na Kiss, Autumn in My Heart, Goong, Full House, and Secret Garden (which is in the works). You'd think they'd do BOF