There are two meanings of the word twee. The sense of twee that we are more familiar with is the more recent and less formal of the two, meaning something a little too cute and sweet. The first recorded use in this sense was in 1905 in the London satire magazine Punch, where it is used as a childish mispronunciation of the word sweet. ("'I call him perfectly twee!' persisted Phyllis." - 1905 Punch 8 Mar. 178/1). This usage has persisted, and while it has drifted slightly from the definition currently found in most dictionaries -- tiny, dainty, miniature -- the sense of 'overly cutesy', as in, for example, the use of baby talk, still remains.
The more formal, but certainly the less common meaning of twee is to refer to 'small instruments', as might be used by a barber who practiced medicine -- things like needles, tweezers, and small knives. Originally, the etweese was the small case containing these instruments (from the French étui, 'small case'), but over time the items inside became known as the tweeze, and, in the singular, twee. Our favorite twee, the tweezers, is currently the only form of this word in popular use. This usage was by far the most common throughout the 1600s, and decreased sharply thereafter.