For nonfiction book translations where the target language does not use the Roman alphabet (e.g. Russian, Indonesian, Mandarin Chinese, etc.), if a particular name, brand, band or song title is mentioned in English, what is the standard approach? Do they stay in English, or should a translator attempt to transliterate them into the target language?
My rule of thumb is not to translate proper nouns unless the translation is already in common use in the target language. For example, it's fine to call Tolstoy's book to "War and Peace" since everyone knows it by that title, but if I started referring to Rio de Janeiro as January River, people would be confused.
Applying this to your case, use a common transliteration if one exists, otherwise stick with the Romanized foreign name and provide a parenthetical translation for clarity as well as to satisfy curiosity.
EDIT: Beyond the above, let common sense guide you. A lot of the time this will come down to style and aesthetics. Sometimes the translation is awkward and so you might prefer the original. And in fiction, you can get a little playful and creative with transliterating with parallel metaphors and suchlike.
Dittos to Cary C.
I'd add that if words in a name are being used as common words and the meanings of those words are relevant, I'd probably translate, while if they are just sounds, I'd transliterate.
To use a Roman alphabet example for simplicity: Suppose I was writing a history book and wanted to mention the German leader, Friedrich der Große. I would almost surely translate this "Frederick the Great". That is, I would translate "der Große" to "the Great", but I would translate/transliterate "Friedrich" to the common English equivalent, "Frederick". The name "Friedrich" comes from German roots meaning "peaceful ruler", but I have never seen an English writer refer to Friedrich der Große as "the Great Peaceful Ruler". Likewise, I have never seen an English writer call him "Frederick the Gross", which would in a sense be a fair transliteration but would completely obscure the meaning of the title.
Likewise, if I was translating a company name that meant, say, "Delicious Fruit Corporation", I'm sure I'd translate that. But if the company used made-up words, I wouldn't try to hack out the root. Like if I was translating "Microsoft" to another language, I'd transliterate, I wouldn't try to translate "micro" and "soft". Even more so if I was translating "Ford": the word "ford" means a place where you can wade across a river, but that has nothing to do with what the company is about, so I wouldn't look for a word in the target language for a river crossing. I'd just transliterate "Ford".
I work on a lot of web sites that get translated into foreign languages, including Far East languages. We usually leave company names in the Roman alphabet, and so will have a string of, say, Japanese characters, than a company name in English, then more Japanese characters. Personally I think this looks very strange, but that's what the translators do. On the other hand, when we have Chinese, Japanese, or Korean companies, for the native language version, of course the name is in, say, Japanese, but for the English version, it gets translated or transliterated into English. I guess far easterners are more accepting of English names stuck in the middle of text in their native language, than English speakers are of the reverse.
In works with the Japanese Language, post WWII, Japanese tend to adopt (American) English words (as well as any other language) pretty handily, and Modern Japanese has many loan words from English from the U.S. Occupation. There is even a format of writing Romance Languages (those that use the Latin Alphabet) into Japanese characters and back based on the sounds made in a Romance Language. Japanese has less sounds than English, so there are some original Japanese words that can get back translated in a few different ways. For example, the Japanese show "Super Sentai" isn't officially translated for Western Fans and to watch it in the west, you need to find a fan sub of the series. One season (Ressha Sentai TOQger) included among the cast a female villain who, depending on the particular fan sub group, translated her name into Gritta/Grita, Glitta/Glitter, or Rita. The last one becoming rather funny given that Super Sentai will lend it's costume hero fight footage to the U.S. show Power Rangers, which also had a villain named Rita. Both also have romantic feelings for a male villain Zedd (or Zetto). Either way, this is because several sounds are used to cover the sounds that do not exist in Japanese and thus can be pronounced multiple ways.
Most English origin loan words are Japaninzed to better fit the languages other characteristics but are not unreasonable for an English speaker to pick out.