Weirdness rhymes with:
hardness, awkwardness, backwardness, badness, baldness, blandness, blessedness, blindness, bloatedness, boldness, coldness, crookedness, discontentedness, fadness, farsightedness, fondness, goodness, handedness, hardness, high-spiritedness, kindness, lightheadedness, low-spiritedness, madness, mindedness, nearsightedness, poor-spiritedness, preparedness, public-spiritedness, redness, rudeness, sadness, sandness, shrewdness, soundness, spiritedness, vividness, wickedness, wildness
Weirdness sounds like:
warden's, wardens, wardens', warding, wartimes, weirton's, wharton's, wording, worthen's, worthiness, worthing, writedowns, writhing, writing, writings