As your question implies, "weed" isn't an inherent biological category, it's a cultural one. However dandelions have some features that help them become weeds. They produce a large number of seeds, which are easily distributed, they grow quickly, they're fairly hardy and they withstand being trampled. They can be outcompeted when other plants become established, but they thrive in opened or disturbed areas.

Those characteristics mean they grow very well in areas disturbed by human activity such as fields and gardens. As long as you want something else growing in these areas - wheat, vegetables, a neat lawn - this makes them weeds. Historically, this sort of bitter wild green was so widely available there was rarely any real need to deliberately grow it. So, while I'll admit I don't have any specific reference to the first time dandelions were recognised as weeds, I'm willing to bet it's as old as agriculture in its native regions.

If anyone happens to know the history of the lawn, that would be a great addition here...

(edit: have recently read Ecological Imperialism by Alfred W Crosby, which talks a lot about weeds.)