Let me try:
Banský is indeed an adjective meaning 'mining'.
I'm unsure about the 2nd parts, but this may be something:
bystriť: to sharpen (a man's name here is Bystrík)
štiav: (bot.) dock, sorrel
So the two Banská cities may owe their names to these two words somehow.
Concerning Bratislava, the Wikipedia has this:
The first written reference (as Brezalauspurc, another variant is Preslavaspurc) comes from the Salzburg Annals, in relation to the battles between the Bavarians and the Hungarians, fought outside the walls of Bratislava Castle in 907. The castle was probably named after Predslav, third son of King Svätopluk I; however, Brezalauspurc literally means "Braslav's Castle" and therefore the town was probably named after Braslav, the last military commander of Pannonia, a province in East Francia. This ancient name reappears under the variant Braslava or Preslava on coins minted by King Stephen I of Hungary and dated around 1000, but the "RESLAVVA CIV" inscription of the obviously conterfeit coins (found only in Sweden) cannot prove the existence of such a name of the town. Later in the Middle Ages, it found its final form in the German name Pressburg and the Slovak name Prešporok derived from it. Pressburg was used to refer to the city by most English-speaking writers until 1919, and it is occasionally so used even today. The Hungarians used Pozsony (spelled Posony before the 19th century) and it is still in use in Hungarian. The Latin name Posonium is derived from the Hungarian. Its Hungarian and Latin denominations might have come from the Hungarian name Poson. In addition to these names, Renaissance documents called the city Istropolis, meaning "Danube City" in Ancient Greek (for example, see Universitas Istropolitana).
The current name, Bratislava, has its beginnings in 1837, when Slavist scholar Pavel Jozef Šafárik invented a variant of it (Břetislaw) from old names, believing that they were derived from that of Bohemian ruler Bretislav I. The name was used for the first time by members of the Slovak movement in 1844 as Bratislav. After World War I, it was proposed to rename the city "Wilson City" (Wilsonovo mesto) after American president Woodrow Wilson. The proposal was rejected, and the official name of the city was changed to Bratislava in March 1919, after the city became part of Czechoslovakia.
So it's apparently named after Svätopluk I's 3rd son, Predslav. Svätopluk was king of Great Moravia from 871-894.
And the Wikipedia has this about Košice:
Wikipedia wrote:The first written mention of the city was in 1230 as "villa Cassa". The Slovak name of the city comes from the Slavic personal name "Koša" with the patronymic slavic suffix "-ice". Though according to other sources the city name probably stems from an ancient Hungarian first name which begins with "Ko" such as Kokos-Kakas, Kolumbán-Kálmán, Kopov-Kopó. Historically, the city has been known as Kaschau in German, Kassa in Hungarian, Cassovia or Caschovia in Latin, Cassovie in French, Caşovia in Romanian, Кошицы (Koshitsy) in Russian and Koszyce in Polish (see here for more languages).
Gløgt er gestsins eyga. (Føroyskt orðafelli)
Wise is the stranger's eye. (Faroese saying)
L'occhio dell'ospite è acuto. (Proverbio faroico)
Hosťovo oko je múdre. (Faerské uslovie)
Fluent: Faroese, Danish, English, German
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