I came across the phrase 甘いものに目がない which roughly translates to "having a sweet tooth". Then I wondered why 目がない refers to "something that you like". Is there an explanation for why "having no eyes" means to "like something"?
Tsuyoshi Ito already touches on the answer. Basically, 目 is used in a lot of expressions talking about your capability see something for its true self (見極める力). It can be seen in phrases and words like: 抜け目がない、目がきく、見識、目角が強い, etc.
So, 目がない literally means you lack the "eyes" to see through things or see something for its true self. Which in turn became to mean that you like something without even thinking or using your ability to see what something really is.