Luk 6:10 Stretch forth thy hand.
Reach out that hand and arm of yours.
This phrase has a simple meaning when addressed to the crippled man, but it also means to go beyond your abilities in extending help. In terms of the larger discussion about religion and the Sabbath, Christ answers his own question about the priority of doing good in the context of religious tradition. He has said clearly that religious tradition is our servant not our master. It is meant to give us power and ability not to take it away. The parallel verse in Mar 3:5 is identical, while the difference in Mat 12:13 is only the position of the pronoun, "your".
"Stretch forth" is a verb that means "to stretch out," but also means "to offer food", "to prostrate yourself," and "to extend." The Greek word for "stretch" has the same meaning as the word in English with all its related ideas of extending yourself and your abilities. This extension of abilities has the same sense of being work and a struggle.
"Hand" is a noun "the hand" but has a host of meanings in Greek beyond a simple body part. It means "helping another" (like the English "lending a hand") and it means "an act or deed," especially in the sense of going beyond words. Christ used the hand, both in his actions and words, as symbolic of an individual's personal power. Every time he used the word (or used his hands), they were an expression of power. Being "in someone's hands" means being in their power (Mat 17:22). He said that we are better off losing our abilities as symbolized by our hands (Mat 5:30) than misusing them.
The phrase meaning "stretch out your hand" also means to "go beyond words to deeds with your abilities."
Ἔκτεινον (2nd sg aor imperat act) "Stretch forth" is ekteinô, which means "to stretch out", "to offer food", "to prostrate yourself", "to straighten", "spread out," to extend," "spin out, "prolong", "put forth" and, in the passive, "be unfolded", "be smoothed."
τὴν χεῖρά (noun sg fem acc) "Hand" is from cheir which means "the hand and arm," and "with the help of agency of another." Like "hand" in English, it has a lot of meanings including "an act or deed", "a body of people," and the measurement "handful."