IIUC you have in mind the definition of an object's {symmetries} as its stabilizer under some group action on its ambient space. While a clearcut origin may not exist (Grattan-Guinness warns: "One of the pitfalls in doing the history of mathematics is "priorities": of accepting, as valid, questions of the form, "Who was the first to...?""), I'd say Jordan has early approximations to what you want, provided you are willing to replace "function" by "transformation", "substitution", or "motion".
E.g. his Traité des substitutions (1870, p. 50) has:
§V. -- SYMMETRY OF RATIONAL FUNCTIONS.
The link between groups and functions.
60 . Let $F_1$ be an arbitrary rational function of $k$ letters $a, b, c,\dots$; $F_1, F_\alpha, F_\beta$ the $1.2.\dots k$ functions obtained by letting the $1.2.\dots k$ substitutions $1, \alpha, \beta,\dots$ operate on these letters (...) The $M$ substitutions $1,\alpha,\dots$ which don't alter the function $F_1$ obviously form a group, which one can call the function's group.
(I would have thought crystallographic groups were defined as stabilizers before Jordan, but it seems not.)