The term 'Faintest Notions' can be broken down into two words, 'Faintest' and 'Notions'. The first word, 'Faintest', is defined as something that is very weak or slight. The second word, 'Notions', is defined as a vague idea or opinion. Together, the phrase 'Faintest Notions' can be interpreted to mean a very weak or slight idea or opinion. The crossword solution for this term is 'INKLINGS', which is defined as a slight suggestion or hint. Therefore, the phrase 'Faintest Notions' can be interpreted to mean a very slight suggestion or hint.