Middle English seien, from Old English secgan "to utter, inform, speak, tell, relate," from Proto-Germanic *sagjanan "to say" (source also of Old Saxon seggian, Old Norse segja, Danish sige, Old Frisian sedsa, Middle Dutch segghen, Dutch zeggen, Old High German sagen, German sagen "to say").
Watkins has this from a PIE *sokwyo-, from a root *sekw- (3) "to say, utter" (source also of Hittite shakiya- "to declare," Lithuanian sakyti "to say," Old Church Slavonic sociti "to vindicate, show," Old Irish insce "speech," Old Latin inseque "to tell say"). Some further see this as identical to the PIE root *sekw- (1) "to follow" (with semantic development to "see" and then "speak"). But others doubt it and Boutkan gives it "No certain PIE etymology."
The past tense form said developed from Old English segde. Impersonal use (it is said) was in Old English. The notion in shall we say, etc. (1580s) is "suppose, take for granted." On that analogy, impersonal say is used as an introduction word, or parenthetically, with a clause and meaning "suppose, assume" (c. 1600). Its colloquial use as an expression of surprise, etc. is by 1830.
Not attested before 1930 in use with inanimate objects (clocks, signs, etc.) as subjects. You said it! "you're right" is attested by 1919; you can say that again as a phrase expressing agreement is recorded from 1942, American English colloquial. You don't say(so) as an expression of astonishment (often ironic) is attested by 1779. The Society of American Florists' advertising slogan say it with flowers dates to 1918 from it grew other say it with constructions.
1570s, "what someone says," hence "what one has in him to say, a declaration or statement," from say (v.). The Old English noun secge meant "speech."
The meaning "right or authority to be heard in a matter or influence a decision" is from 1610s in have a say; earlier in this sense was have a saying (late 15c.). Extended form say-so "personal assertion" is recorded by 1630s; in the sense of "power, authority" it is by 1896.
"named or mentioned before," c. 1300, past-participle adjective from say (v.). Expression all is said and done is from 1550s.