“gift.” The Hebrew “gift” is used quite a bit with sacrifices. The offering is almost like Passover: a goat and unleavened bread (the Passover lamb could be a goat). There is something going on in Gideon’s mind that he will be more certain that this messenger represents Yahweh if Gideon gets to bring an offering.
“sit.” The Hebrew word usually means “sit,” but it can mean “live” or “dwell,” or also “wait.” In this case, it may mean “wait,” but it could also have its most common meaning of “sit,” and it may indicate that the angel had never gotten up from sitting under the tree (see Judg. 6:11).