Board Thread:Questions and Answers/@comment-108.90.8.74-20150806221957/@comment-1173941-20150806234632

There are hair-dye items. Even Diavel in the second episode used blue hair-dye, even though it should be expensive for people around level 10, just to seem more important.

We're not given a date or time-frame of when Liz got her hair dyed, but according to the light novel, it was Asuna's suggestion, to make her look cuter, because Lisbeth has what Asuna would call a "baby face", basically.