Objection 1. It would seem that the relics of the saints are not to be worshiped at all. For we should avoid doing what may be the occasion of error. But to worship the relics of the dead seems to savor of the error of the Gentiles, who gave honor to dead men. Therefore the relics of the saints are not to be honored.
Objection 2. Further, it seems absurd to venerate what is insensible. But the relics of the saints are insensible. Therefore it is absurd to venerate them.
Objection 3. Further, a dead body is not of the same species as a living body: consequently it does not seem to be identical with it. Therefore, after a saint's death, it seems that his body should not be worshiped.
On the contrary, It is written (De Eccles. Dogm. xl): "We believe that the bodies of the saints, above all the relics of the blessed martyrs, as being the members of Christ, should be worshiped in all sincerity": and further on: "If anyone holds a contrary opinion, he is not accounted a Christian, but a follower of Eunomius and Vigilantius."
I answer that, As Augustine says (De Civ. Dei i, 13): "If a father's coat or ring, or anything else of that kind, is so much more cherished by his children, as love for one's parents is greater, in no way are the bodies themselves to be despised, which are much more intimately and closely united to us than any garment; for they belong to man's very nature." It is clear from this that he who has a certain affection for anyone, venerates whatever of his is left after his death, not only his body and the parts thereof, but even external things, such as his clothes, and such like. Now it is manifest that we should show honor to the saints of God, as being members of Christ, the children and friends of God, and our intercessors. Wherefore in memory of them we ought to honor any relics of theirs in a fitting manner: principally their bodies, which were temples, and organs of the Holy Ghost dwelling and operating in them, and are destined to be likened to the body of Christ by the glory of the Resurrection. Hence God Himself fittingly honors such relics by working miracles at their presence.
Reply to Objection 1. This was the argument of Vigilantius, whose words are quoted by Jerome in the book he wrote against him (ch. ii) as follows: "We see something like a pagan rite introduced under pretext of religion; they worship with kisses I know not what tiny heap of dust in a mean vase surrounded with precious linen." To him Jerome replies (Ep. ad Ripar. cix): "We do not adore, I will not say the relics of the martyrs, but either the sun or the moon or even the angels"--that is to say, with the worship of "latria." "But we honor the martyrs' relics, so that thereby we give honor to Him Whose martyrs [The original meaning of the word 'martyr,' i.e. the Greek martys is 'a witness'] they are: we honor the servants, that the honor shown to them may reflect on their Master." Consequently, by honoring the martyrs' relics we do not fall into the error of the Gentiles, who gave the worship of "latria" to dead men.
Reply to Objection 2. We worship that insensible body, not for its own sake, but for the sake of the soul, which was once united thereto, and now enjoys God; and for God's sake, whose ministers the saints were.
Reply to Objection 3. The dead body of a saint is not identical with that which the saint had during life, on account of the difference of form, viz. the soul: but it is the same by identity of matter, which is destined to be reunited to its form.
Source: Summa Theologica (New Advent)