The Qin Dynasty is remarkable in unifying China for the first time, but this accomplishment alone is not enough to explain the name. Two reasons: one, Qin Dynasty is far from the most remarkable even to Chinese people, who might refer to themselves as 漢人 (Han people) or even 唐人 (Tang people), and never as "Qin people". Second, when it comes to names foreigners generally don't give a damn what your domestic accomplishments are, they just use the first name they hear that sticks. See for example the names of Germany or the American "Indians".
The real reason would be hard to prove, but I think there are two strong reasons why the name came from Qin:
First contact with the Greco-Bactrians
One of the earliest contacts between the West and China might have been during the reign of the Greco-Bactrian king Euthydemus I, who ruled between 230–200 BC. During his reign the kingdom continued its eastward expansion, and may have come in contact with the contemporaneous Qin dynasty (221 to 206 BC). Thus the Qin may have given its name in a tremendous luck of timing.
Qin's position in the West
But even if contact existed, albeit unrecorded, long before this, it's also very likely that Western observers would have encountered the Qin state, which later conquered China and became the Qin dynasty. At the start of the Warring States period (~450 BC), Qin was already a major state in the far West, and became dominant via aggressive westward expansion and opening up lucrative westward trade (its capital Xianyang is basically in the same area as Chang'an, the famous Silk Road city). Anyone wishing to trade with China during this time would basically pass through Qin.