If the Ch’in dynasty was so short-lived, why was China named for it?

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.

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The Ch’in dynasty united modern China and greatly advanced Chinese culture; that's why modern-day China is named for this short but great dynasty.

The remarkable Ch'in dynasty arose from the remains of the Chou dynasty in 255 B.C.

Although it lasted only 40 years, the Ch’ins united almost all of China and also finished constructing the Great Wall.

The Qin dynasty ruled in China from 221 BC to 206 BC. It was created by the warlord Qin Shi Huang during the Warring States Period and defeated several other states in the area to unite China. While the Qin ruled China for only a very short time, they built both the Terracotta Army and parts of the Great Wall of China. The second, and final emperor, Qin Er Shi, was overthrown by a popular rebellion after the first Qin emperor died. It was followed by the Han dynasty. The Han dynasty was founded by Liu Bang in 202 BC.

Source: Simple English Wikipedia

The Chinese actually refer to themselves as the Han most of the time. Why do westerners refer to China as such then? Well, it is more complicated than you might think. China is either a Persian or Sanskrit word for a kingdom in central Asia that might be the Qin dynasty, or might be something else. Marco Polo seems to be the one who brought this name back to Europe after traveling to both these places, but where he got it from exactly is a mystery.

The original question is a bit vague as to who exactly is calling China, well, "China." This also means that it's a bit tricky to pin down what exactly the word "China" refers to (e.g. is it the same word as "Cathay," "Sina," etc.? Is it even referring to a nation?), especially because it's not at all a certainty that "China" is derived from "Ch'in/Qin" (I'll refer to it by its Pinyin romanization "Qin" for the rest of this answer). It's also not exactly clear what "Qin" refers to as there are many political entities in China's history that shared that name. The first dynasty to unify China? The state that preceded the dynasty? One of quite a few states in later periods of disunity? So let's address each part of that.

1. Do Chinese people now refer to, or have they ever (other than during the Qin Dynasty or the rule of a state named Qin) referred to their nation as some derivative of the Chinese character "Qin" (秦) of the Qin Dynasty? No (except for one possible round-about exception, "Zhina," which is today viewed as an offensive slur). Let's leave aside who exactly "Chinese" refers to unless significant confusion in the comments arises. That's a whole 'nother can of worms.
2. Are the names that China has been referred to through different times and different regions etymologically related to the English word "China?" Many are, but far from all.
3. Is there some etymological connection between the English word "China" and the Chinese character "Qin?" There seems to be a general academic trend towards a weak "Yes" tempered by significant dissent.
4. If the answer to 3. is yes, what political entity are we talking about? Probably the first dynasty or the state preceding it in the Warring States period.
5. If the answer to 3. is yes, is there any particular reason that name was chosen? These kinds of "why?" questions are almost impossible to really answer well and/or rigorously, but the answer is probably because that just so happened to be the name of the state in power when contact was made.

Here's some more details on each one in turn. But first caveats galore.

• Literal translations of names are misleading: When I say the name "Peter Smith," am I implicitly thinking of "Stone Blacksmith" or am I just thinking of a name with no other connotation? Probably the latter. Offering literal translations often gives an analogous mistaken impression of additional semantic meaning that just doesn't exist ("Peter Smith, man that guy's probably a stony fellow who's great at shaping metal!" is probably not what you're thinking of right now). So take my literal translations with a grain of salt. It's often difficult to disentangle when the original text meant to evoke the meaning of the words themselves in the name and when the words are used simply as a proper noun.

• Nailing down exactly what a "nation" (and hence what actually corresponds to the notion of "China") is gets pretty difficult as you go back in time: Modern nation-states have a tendency to tie together a piece of land, a government institution, and a cultural heritage in a single package. These things historically have often not been packaged together in many societies and "China" is no exception. A single government might control many different lands and many different peoples, a single land might be controlled by many different governments and many different people, or a single people might be subjects of many different governments and live in many different lands.

  I'm going to be moving loosely between words that apply to land that is a sizable subset of modern-day China and people whose culture most Chinese people would probably agree is part of their cultural heritage.

With those in mind, let's dive in!

1. The Chinese people have called their nation and themselves many names through time, many of which come from dynastic names. Qin does not figure among them however (except during times and places of rule by the Qin). The first dynasty of China had a rather bad rap among later generations, mainly due to massacring scholars who didn't follow the Legalist school of thought. As the early 20th century Sinologist Berthold Laufer put it:

   [The] Chinese people never called themselves after the Ts'in for whom their scholars professed a thorough contempt... (The Name China. Berthold Laufer. T'oung Pao, Second Series, Vol. 13, No. 5 (1912). pg. 723)

   Here's a list of alternatives.

   • Dynastic ones:
     • Xia (夏): Perhaps semi-mythical early dynasty of China. As a name for China or the Chinese people often shows up in the form 华夏/華夏 ("Glorious Xia"). The first part of "Glorious" (华/華) has since had its meaning shift over time such that even in isolation that word means Chinese (see for example 华人, a synonym for "Chinese people").
     • Han (汉/漢): Dynasty immediately succeeding the Qin. Alongside 华/華 probably the most used current endonym with dynastic origin by the Chinese. Popular phrases include the Chinese language (漢語), Chinese characters (漢字), and the majority ethnicity in China (漢族).
     • Tang (唐): In modern parlance shows up in Southern Chinese vocabulary in such phrases as one Chinese name (among several) for foreign Chinatowns (唐人街 "Tang People St./The street of the Tang people.").
   • A small selection of some non-dynastic ones:
     • Middle INSERT_PHRASE_HERE:
       • Middle Capital/State/Kingdom (中國): Current most popular name for China and the one with the longest history, stretching back to the 11th century BCE in the Western Zhou dynasty. Its meaning has varied over time. For much of ancient Chinese history it has been used to refer to the Central Plain area and more broadly used as a metonym for the culture and institutions that are traditionally identified as originating from there. The Warring States Period sees this term often used just to describe the different capitals of each state at the time. The notion of a political entity calling explicitly itself 中國 sees some echoes in the Ming Dynasty (the few Ming examples I see make it somewhat unclear whether once again these are metonyms or actual names that a government gives itself) and is definitely confirmed in the Qing Dynasty.
       • Middle Ground (中土): See Central Plain.
       • Middle Plains/Central Plain (中原): The Central Plain region of China (an area corresponding to a section surrounding the Yellow River) was the seat of many of its early civilizations (e.g. the Shang and Zhou dynasties) and hence often identified in ancient texts often in contrast to other foreigners.
       • Middle Glorious/China China (中華): An amalgamation of 中國 and 華夏. Also popular today and figures in both the current name of the People's Republic of China (中华人民共和国) and the Republic of China (中華民國). While literally "Middle Glorious," each character now in modern usage independently represents a synonym for "China" or "Chinese."
     • Nine Provinces (九州): An ancient name for China. What exactly the nine provinces were is, amusingly, not exactly clear. It seems every ancient Chinese source that mentions them lists a different nine.
     • Zhina (支那): A term for China that shows up occasionally in Chinese translations of Sanskrit texts; historically had a small amount of usage within Chinese Buddhist circles. Derived from Sanskrit later usage of cina (in turn possibly derived from Qin, making this an interesting reborrowing back into Chinese). Enjoyed a brief, weak resurgence at the turn of the 20th century among some revolutionary Chinese intellectuals looking to distance themselves from the Qing/清 (not Qin/秦) Dynasty, when reimported from Japan. Quickly became a slur in the wake of the Japanese occupation of China.
2. To give a flavor of the various exonyms China has had, here's a selection of ones both etymologically related and not related to "China." I'll start off by noting that "Qin people" (秦人) is an exonym recorded in usage by northern civilizations in contact with the Chinese, although it's not quite settled whether that it made its way over to Indo-Europoean languages.
   • (Probably) Related to "China":
     • The mass of Indo-European languages: Sanskrit/Roman/Greek/French/Spanish/etc. (Cina/Sīnae/Sînai/Chine/China respectively): The first three are kind of tricky, because it's not clear they knew who they were talking about. Most of the time the word just meant "a bunch of people to the East/North" with a smattering of tantalizing clues (e.g. "they have silk," "they're over the ocean," "they're at the edge of the world").
     • Pre-WWII Japan (shina/支那): See "Zhina."
   • Not related to "China"
     • Russian (kitaiski/китайский/"Chinese"): Derived from the Khitai (契丹), the name of a group who lived in what is modern-day northern China.
     • English "Cathay:" Also derived from Khitai.

3. Early 20th century Sinologists such as Paul Pelliot and Berthold Laufer expounded at length about the likelihood that the Indo-European cognates of "China" all come from Qin (Laufer in particular changes his opinion later in life from initially opposing the idea to coming around to it). This is far from unanimous though. Endymion Wilkinson instead seems to attribute the origin to "silk":

   "Seres" (the silk people) may have been intended by some Roman writers to refer ot the Chinese. "Silk" possibly came from Latin sericus, itself in turn derived by way of Greek and Roman from the Old Chinese pronunciation sie for si (Chinese History: A Manual. Wilkinson. Pg. 752)

4. This mainly matches with the timelines established for most of the Indo-European cognates claimed to have descended from Qin. In particular, Laufer and Pelliot both mention a 300 B.C. dating of the Sanskrit word Cina given by Hermann Jacobi, which Laufer initially takes as evidence against the Qin origin and which Pelliot seems to hand-wave away as a potential hedge that perhaps "China" comes from the pre-unification Qin state, but still believes

   L'opinion traditionnelle, qui invoque le souvenir de Ts'in Che-houang-ti, me paraît encore, sinon certaine, du moins le plus probable. (L'Origine du nom de "Chine". Paul Pelliot. T'oung Pao, Second Series, Vol. 13, No. 5 (1912). Pg. 740.)

   The traditional opinion [that "Chine" comes from Qin], which invokes the memory of Qin Shi Huangdi, still appears to me, if not certain, at least the most probable.

5. At the end of the day it seems like all the academic sources just settle on Occam's Razor. Like I said, hard to answer well or rigorously.