Through him therefore let us offer the sacrifice (or, a sacrifice) of praise to God continually, that is, the fruit of lips confessing to his Name. θυσία αἰνέσεως is the designation in the ritual of the Law of the voluntary peace offering, offered by individuals on occasions calling for special thanksgiving (Le Hebrews 7:12).
In the psalms it is used to express generally praise and thanksgiving (see Psalms 1:1-6:14, 23; Psalms 116:17. θῦσον τῷ θεῷ θυσίαν αἰνέσεως καὶ ἀπόδος τῷ ὑψίστῳ τὰς εὐχάς σου, etc).
In virtue of their participation in the true and complete Sin Offering, Christians may fulfill this part of the ancient symbolism, not occasionally, but "continually;" bringing to God, not fruits of the earth, but the "fruit of the lips" (an expression found in Hosea 14:2, where the LXX.
has καρπὸν χειλέων ἡμῶν), i.e. continual praise, springing from thankful hearts. In the Eucharist especially (hence so called) such sacrifice is continually offered, over the one atoning Sacrifice which is pleaded and partaken of.
But not in communions only, but ever in their daily lives, such "sacrifice of praise and thanksgiving" is due. But, as the next verse reminds the readers, the "knit of the lips" is not enough; there is a further sacrifice of our own, whereby we must show that we are true partakers of Christ, and truly thankful.