I've always felt that tit is not particularly offensive. Not, at least, in a context like:
Oh, you silly tit!
I was attempting to demonstrate to somebody that the word is not offensive, at least not in BrE, and had assumed that the etymology would back me up. I was guessing it would have more to do with small birds than with teats. However, the Online Etymological Dictionary fails me here:
tit (n.2)
1540s, a word used for any small animal or object (as in compound forms such as titmouse, tomtit, etc.); also used of small horses. Similar words in related senses are found in Scandinavian (Icelandic tittr, Norwegian tita "a little bird"), but the connection and origin are obscure; perhaps, as OED suggests, the word is merely suggestive of something small. Used figuratively of persons after 1734, but earlier for "a girl or young woman" (1590s), often in deprecatory sense of "a hussy, minx."
That doesn't really sound like the tit of you silly tit. The OED is better but not very illuminating either:
tit, sb.7 slang. Etymology: Of uncertain origin: perh. f. tit sb.6; cf. tit sb.3, twit sb.1 2 b.
A foolish or ineffectual person, a nincompoop.
So, in the unlikely event that someone is better informed than the OED, what is the etymology of this sense of the word tit? Is it related to the birds? To teats? To something altogether different?