Meaning 'the best'- The handkerchief that is better than all the rest - usually with a strong attractive pattern / design that makes it the 'Hankiest if Hankerchiefs'
Bob - "Is that blue and burgandy striped handkerchief yours?"
Derek - "Yeah, it's my favourite"
Bob - "Well it sure is a nice looking one"
Derek "It's the Hankiest of Handkerchiefs"
to be the highest degree of something classified as honkey...the word "honkey" plus the superlative adjective "est" combine to form a degree of honkeyness rarely seen in urban culture
"What's up with all the HILLYBILLY HANKIES? These people know the Civil War's over, right?" -Dean Winchester, Supernatural, Season 8, Episode 6, "Southern Comfort"