Height is the only correct version of this word today. Heighth is no longer considered a standard spelling of this word, if it ever was. Since height rhymes with flight and might, two other words that end in -ght, you will always know how to spell the ending of the word.
What does heighth mean?
1a : the part that rises or extends upward the greatest distance : the highest part : summit reached the height of the mountain.
Why do we say heighth?
The word began in Old English with a –th sound near the end, analogous to width, length, depth and breadth. It was spelled híehþo – that þ letter is a thorn and signifies a voiceless dental fricative: the 'th' sound in heighth.