"Inklings, like tinklings, were small sounds. Specifically, an inkling was an indistinct hearing of the use of one's own name. This meaning is demonstrated in the earliest known use of the word, in the Alliterative Romance of Alexander the Great, circa 1400–50: (Alexander) Herd a nyngkiling of his name, & naytis him (exerted himself) to ryse.
So, if you heard an inkling your senses were alerted to listening for something that might interest you and it is that sensation that has lead to our current usage of the word. To be more accurate, in the 13th century you would have had a ninkling rather than an inkling. Inkling is one of those nouns that were originally spelled with an 'n' at the start of the word but later lost it in everyday speech. In this case, sometime in the 16th century, 'a ninkling' became 'an inkling'. Other examples of this are 'a napron' and 'a nadder' snake. This reformation of words is called metanalysis and these 'n' examples of it are difficult to explain fully." - phrasefinder.co.uk