What does Shivansh mean?
A modern compound of Shiva and amsha (अंश, 'portion/part'), read as 'a part of Lord Shiva'. A contemporary devotional coinage rather than a classically attested name.
What does Neel mean?
From the Sanskrit नील (nīla), 'dark blue, indigo, sapphire'. A genuine classical colour word, also the root of 'neelam' (sapphire). Note: the similar-looking English/Irish name Neil (from Gaelic Niall) is a separate, unrelated name.