Ling is being used as a diminutive suffix but the ink comes from incen, meaning to mutter or whisper. So an inkling is a soft whisper.
However, the word has suffered every indignity at the hands of lexicographers and a giddy public. Dr Johnson, wrongly but reasonably, thought that it was to do with clink. He wrote that "The word is derived by Skinner, from inklincken, to sound within", a bit like tinkling.
Others thought that it was related to inclination, which is why Robert Southey wrote in 1824 "I feel inklings to address an ode to the people of Liverpool", a usage as perverse as the desire that it expresses.
Still others decided (as with sideling) that there simply must be a verb inkle of which inkling was the gerund. This seems to have been employed mainly by the pseudo-dialect school of writing. Richard Blackmore, who wrote Lorna Doone, has "His marriage settlement and its effects, they could only inkle of", Hardy uses it to typically tedious effect, but Samuel Butler in Erewhon Revisited has the lovely line:
People like being deceived, but they also like to have an inkling of their own deception, and you never inkle them.