The name 'Danube' is said to derive from Proto-Indo-European, a language reconstructed from European & Indic languages. In PIE, the word 'dānu' is apparently a term for "river", but PIE of course does not have any scriptures or literature that detail the context in which the Danube was called "danu". The word "Danu" simply translates as "a river". It is said that the PIE root word is 'da' which means 'rapid, swift and violent'.*
In Sanskrit, the word 'Danu" (दानु) has many meanings. It means 'dew, dew drops, fluid, valiant and courageous'. The root word is 'da' (दा) which means 'that which is cleansing and purifying, giving and protecting'. This is probably more appropriate as a name source for a revered river.
The Rig Veda and Puranas identify the details of Goddess Danu and her sons, the Danavas. In the Vedic context, 'Danu' was a primeval cosmic river. Goddess Danu, the daughter of Daksha, who was the son of Brahma, embodied the primeval waters. For more details about the story of Goddess Danu and Danavas click here.
*Just a quick look at other related Sanskrit words. 'Drav' (द्राव) means 'going quickly' and (द्रुत) means to run. The root word here 'dru' (द्रु); which means to 'run', 'go swiftly', 'melt', 'dissove' and 'become fluid'. The 'dru' root word may describe the river's swiftness, but it lacks the 'cleansing and purifying, giving and protecting' aspect that the 'da' (दा) root word adds to the description of the river.
The Sanskrit word for 'rapid' is "Ishiram" (इषिरम्) which is the etymological source for "Istros", the Greek name for the River Danube.