There are not many instances of -eigh- or -aigh- words: eight, height, neighbor, weigh, straight are the most common. Height in particular had two variant pronunciations going back to Middle English, yielding differing pronunciations in the early Modern period:

The lower variant in the table (beginning with the long [i:] in late Middle English) is the one that won out with height. With the other words listed (eight, neighbor, weigh, straight) the upper variant was followed exclusively, yielding the pronunciation we have today.