The five answers given quote the facts, but I’m afraid they don’t understand the facts.
Nigh comes from the original Old English word for “near”.
Near comes from the Old Norse for “nearer”. It came to England with the Vikings.
They are not the same etymology. They are related (cognate) words, just as shirt from Old English and skirt from Old Norse are related: but the last time they constituted the selfsame word (ignoring that one is a comparative) was in proto-Germanic. In 800, when the Vikings came to England, English nēah and Norse nær were two separate words from two separate languages.
My thanks to Syarif Fadhlurrahman for his clarification in comments.
Near comes from Old English with some influence from Old Norse. It’s not totally from Old Norse:
Granted, Oxford assigns only the Old Norse etymology, but I don’t see why. Perhaps due to non-adjectival use? (Old Norse ‘naer’ can function as adverb and preposition)
, MA in Linguistics from BYU, 8 years working in research for language pedagogy. and
, Linguistics PhD candidate at Edinburgh. Has lived in USA, Sweden, Italy, UK.