This appears to be slang, concentrated mostly in the Southern US. It is widely reported there, particularly in the Carolinas and Georgia, and its definition is just as you have it.
For example, from the blog The Room Mom:
What is a Sirsee?
Spelling variations: circe, circi, surcy, surcee
Definition (n): word used in the South to mean a small, thoughtful gift, unexpected gift, just because.
Because the word is mostly used in informal, oral, contexts, there's not a lot of literary history or evidence, so its orthography is not standardized, and its origins are murky and uncertain.
So, of course, the Internet abounds with speculation. Here's some of the more credible speculation, from the Word Detective:
Do you know where the word “sercy“, “surcy“, “cerci” or “searcy” comes from? Do you have a plausible theory? Do you use it in conversation?
Word Detective posits:
I posed the question to the American Dialect Society mailing list, and received the following reply from Joan Hall, Associate Editor of DARE (Dictionary of American Regional English):
“The Dictionary of American Regional English files have anecdotal evidence for the term “sirsee”¹ (variously spelled “circe,” “circi,” “surcy”) from NC, SC, GA, and PA, as well as two reports from Buffalo, NY and Oklahoma, where the speakers were said to come from “someplace else.”
“All evidence is oral, so the spellings are speakers’ attempts to represent the pronunciation. The etymology is uncertain, but one plausible source is the Scots verb “sussie,” meaning “to take trouble, to care, to bother oneself.” This probably came to Scotland from the French “souci,” meaning “care, trouble.” The term seems to be especially well known in Columbia, SC, where it is strongly associated with a women’s college. Michael Montgomery of the English Department of the University of SC has reported its use there.”
Furthermore, as @Frank pointed out in the comments, a couple of Southern businesses, Surcees.com, a Southern florist, and Charleston Surcees, a caterer, have embedded the word surcee (so spelled) in their names, and they echo the potential etymology based in Scottish and Irish loanwords:
Surcee is a word, used in the South, that is believed to have its origins in the Scotch and Irish who settled in the region. The Scottish word for "surprise" is pronounced much like "surcee" and could be a phonetic form passed down from generation to generation. Another possible origin for the word is from the Irish term "sussie" which means to care.
I happened to be in Charleston the other week, and passed by Charleston Surcees, so I took a couple of pictures of the word used in the wild:
And one of a collection of various surcees, laid out beneath their definition:
All sources emphasize the word is uniquely Southern.
¹ As indicated, the Dictionary of American Regional English (DARE) does indeed have a headword for "sirsee", but unfortunately one needs a subscription to access its definition and details.