The idea of bones (relics) having some supernatural power probably sources to the story of the dead man touching the bones of Elisha and returning to life.
And it came to pass, as they were burying a man, that, behold, they spied a band of men; and they cast the man into the sepulchre of Elisha: and when the man was let down, and touched the bones of Elisha, he revived, and stood up on his feet.
2 Ki 13:21
In addition, the due respect of the dead would source back to Joseph's request when the Israelites left Egypt.
And Joseph said unto his brethren, I die: and God will surely visit you, and bring you out of this land unto the land which he sware to Abraham, to Isaac, and to Jacob. And Joseph took an oath of the children of Israel, saying, God will surely visit you, and ye shall carry up my bones from hence.
This great example of faith in God's promises was carried out by Moses and mentioned in the list of faith examples in Hebrews 11.
And Moses took the bones of Joseph with him: for he had straitly sworn the children of Israel, saying, God will surely visit you; and ye shall carry up my bones away hence with you.
By faith Joseph, when he died, made mention of the departing of the children of Israel; and gave commandment concerning his bones.
In those examples, it wasn't so much that the bones had some mystical power, but rather they represented the reality of God's promises.
At the time of Christ, they no more would revere bones, than touch them. It was against the Law of Moses.
He that toucheth the dead body of any man shall be unclean seven days.
And Christ will contrast the idea of clean on the outside, while dead on the inside.
Woe unto you, scribes and Pharisees, hypocrites! for ye are like unto whited sepulchres, which indeed appear beautiful outward, but are within full of dead men's bones, and of all uncleanness.
So to answer the OP, there is no belief in the power of bones at Christ's time and no indication of Christians who were revering inanimate objects.
But subequent to the ascension some hundreds of years later, like with Elisha, there were factions that again saw bones (relics) as somehow useful to "hear" God. The earliest account, not of relic gathering, but of recognizing God's promises, is written in the Martrydom of Polycarp circa 170.
The centurion then, seeing the strife excited by the Jews, placed the body [of Polycarp] in the midst of the fire, and consumed it. Accordingly, we afterwards took up his bones, as being more precious than the most exquisite jewels, and more purified than gold, and deposited them in a fitting place, whither, being gathered together, as opportunity is allowed us, with joy and rejoicing, the Lord shall grant us to celebrate the anniversary of his martyrdom, both in memory of those who have already finished their course, and for the exercising and preparation of those yet to walk in their steps.
As well there is the case of the marytrs of Lyons, such as Blandina.
The heathen were not content with putting the martyrs to death with tortures, or allowing them to die in prison. They cast their dead bodies to the dogs, and caused them to be watched day and night, lest the other Christians should give them burial; and after this, they burnt the bones, and threw the ashes of them into the river Rhone, by way of mocking at the notion of a resurrection. For, as St. Paul had found at Athens (Acts xvii. 32), and elsewhere, there was no part of the Gospel which the heathen in general thought so hard to believe as the doctrine that that which is “sown in corruption” shall hereafter be “raised in incorruption;” that that which “is sown a natural body” will one day be “raised a spiritual body”
The idea of burning a Christian's body so as to leave nothing for God to work with is laughable, but unfortunately some took that attitude too far in the opposite direction that God would work bones or inanimate objects such as thorns, a cup, a piece of wood for the living.
In any event, there's no sense of the idea that came to grow later in the church of some latent power in relics, but rather just a keen awareness of God's promises and faithful people.