The short answer is that he reigned a really long time, expanded the Empire to its greatest extent, and we have some interesting sources from the period.
He was emperor from 527 - 565 CE, and had the remarkable good fortune to have a fully stocked treasury, courtesy of the emperor Anastasius, competent subordinates, and the opportunity to reshape Constantinople. All of this allowed him to engage in costly, but mostly futile, foreign conquests, and commission incredible building projects like the Hagia Sophia.
Additionally we are able to get a pretty good sense of the man from the writings of Procopius of Caesaria by contrasting what he wrote in his very flattering official histories against his scathing secret histories. The mural of him in Ravenna also helps keep him relevant because it is one of the few portraits we have of a Byzantine emperor.
He is far from the only emperor of note in the thousand years between the fall of the western empire and the fall of Constantinople, but he is one of the most visible. In a similar vein Julius Caesar or Augustus weren't the only Roman emperors of note, but they are the most visible.
This might not be so relevant for common-law countries but one important reason is Justinian's role in the compilation and promulgation of the Digest, Code, Institutes plus some other "constitutions" later called the Novels. They formed the basis for the academic study of law in Western Europe from the 11th century onward and later for the actual law in the courts and legislation. These compilations were in Latin or translated into Latin in the case of the Novels and so they were comprehensible to the Latin-reading parts of Europe. Later Byzantine legislation like the Ecloga were in Greek and came at a time when the empire was losing it's connection to the Latin West, so to speak. Roman law as known later in Western Europe was defined by Justinian's activities.
If anyone's curious about Justinian, or just the history of the Byzantine Empire, I highly recommend the History of Byzantium Podcast. He outlines in the episodes concerning Justinian, why he's so popular, what he did for the empire, and whether he really deserves the hype.
As the first part of your question has been more than sufficiently answered, I shall endeavor to answer the second with the tale of another Emperor who ruled not long, relatively, after Justinian. This was the Emperor Heraclius who ruled from 610-641 during one of the most important time periods to ever shape the Middle East.
To understand why Heraclius was of import equal to the lauded Justinian we must first put into place some context. After the death of Justinian his nephew took the purple and proceeded to cause, at the risk of a massive over-generalization, havoc throughout the empire his uncle had so recently conquered both militarily and religiously, then after starting a war with Persia he proceeded to go insane (This is a massive summary of titanic events)(Treadgold 223). The Crown then fell to one Tiberius, who proceeded to burn through the treasury that Justin had accumulated and fought a bloody, costly, and inconclusive war with the Persians. Barely able to maintain hold of both Justinians reconquered west and the lucrative East, as well as fending off the Avar threats from the North Tiberius died leaving the Empire a little stronger, but considerably poorer (Treadgold, 227).
Tiberius was followed by Maurice who had to deal with the three major fronts left by Tiberius. Unable to decisively win on any front and with his treasury rapidly dwindling Maurice tried to reduce the pay of the army by issuing the soldiers arms directly instead of paying the troops to buy their own equipment (Treadgold, 229). This resulted in immediate mutiny of the eastern armies who then proceeded to loot the surrounding region. The mutiny was eventually quelled in 589 and the happily for the Byzantines one of the Persian generals revolted against the King, unhappily the now deposed king of Persia asked the Byzantines to place him back on the Throne. Maurice agreed and succeeded and for his aid the Persians returned all of the land they had seized during the previous decades of fighting. Maurice slowly restored the status quo, but was overthrown by the Usurper Phocas.
Maurice had saved an empire that had all of the signs of crumbling, when he was deposed in 602 the treasury was bare, but the empire was in no major conflicts and all could have been set to rights if anyone other than Phocas had been at the helm. With the accession of Phocas the eastern armies revolted and the restored King of Persia declared what would end up being a long and bloody war. At this point, with famine wracking the empire an empty treasury and over half of the empire in revolt [Africa, Egypt, the Eastern Armies] (Treadgold, 241) with Persians pushing deeply into the heartlands of Anatolia and with Slavs burning their way across Thrace and Macedonia Heraclius, himself part of the African revolt, sailed to Constantinople and ended the civil war by executing Phocas.
At this point in time the Empire was extremely close to complete and total financial, political, and military collapse (Treadgold, 241). Heraclius himself came to the throne with very little military experience although he would soon show himself to be the military genius that the empire needed. With inconclusive fighting in the years following his accession Heraclius took control of a large portion of the Empires remaining military strength in 613 (Treadgold 289). By 615 the empire had lost control of nearly all of [Byzantine] Mesopotamia, Syria, Palestine, and Cilicia to the Persians, Illyricum to the Slavs, and Spain to the Visigoths. The only, relatively, intact territories of the empire were Africa, Egypt, and Anatolia the latter two were also facing imminent Persian invasion. Egypt fell to the Persians with little resistance and the Persians fully conquered it by 620. The empire at this point was a rump state that could barely pay even half of its army and held control of Greece and parts of Anatolia, as well as distant Africa which was to far to truly be of assistance. The emperor at this point took drastic action and through a combination of careful planning, total restructuring of the military, and melting down the gold of the churches, (with the Patriarch's consent) managed to push the Persians out of Anatolia with one of the first victories over the Persians in years (Treadgold 293-4). The Emperor led, comparatively, a lightning thrust into the contested lands in Armenia, seemingly in an attempt to drive his way toward Ctesiphon, which was the Persian capitol (Treadgold 295-7). The Persians responded in kind and in 626 had driven all the way to Chalcedon just across the Bosphorus from Constantinople, at the same time Persian allied Avars cut the Aqueduct of Valens and began raiding the suburbs outside of the walls. The Persians were ferried across the strait and were promptly routed and the Avars were driven back across the Danube. This is where the tide in this titanic struggle between ancient Empires truly turns in the favor of the Byzantines (Treadgold 298-300). From there Heraclius burned and pillaged his way across the heartland of Persia and advanced upon the Ctesiphon itself. in the year 628 the struggle between Persia and Byzantium ended, with the restoration of the borders as they had been in 602. Heraclius then defeated a rogue Persian General and seized the True Cross, which had been taken from Jerusalem in the opening stages of the war, and returned it to that holy city. In the year 631, Heraclius finally set down his sword to begin governing the Empire that he had been fighting for his entire reign, one year before the Prophet Muhammad died and the Islamic explosion began. By 641, The year the emperor died in bed, everything he had worked to hold onto had been lost to Islam. The byzantine exarchate of Africa, the Emperors homeland, managed to hold on until being crushed in 670, but the Empire would never return to the lands that had been bled for for so many years.
Treadgold, W. T. (1997). A history of the Byzantine state and society. Stanford University Press.