Was the Gathering Relics of Holy Men in 1st Century common in the Early Church?

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. Gen 50:24-25

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. Exo 13:19

By faith Joseph, when he died, made mention of the departing of the children of Israel; and gave commandment concerning his bones. Heb 11:22

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. Num 19:11

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. Mat 23:27

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. -source-

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” -source-

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.

Neither Britannica nor Wikipedia list any form of relics within Judaism.

Collecting relics was effectively never done by the Israelites or Jews. To them, only God (and God through his agents) has supernatural power. Believing that relics or other objects have supernatural power, in principle, violates the first and second commandments, "Thou shalt have no other gods before me" and "Thou shalt not make unto thee any graven image". It in effect turns the object into a form of god.

Ancient Greeks though, did believe in the power of relics, and this belief passed on to the Roman Empire and the Holy Roman Empire.

It wasn't until the Roman Church became the official Christian form of Christianity that anyone thought to collect Christian relics.

Constantine's mother, Empress Helena, was particularly interested in relics, artefacts, and locations. She herself made a pilgrimage to Palestine and other provinces where she built churches on holy sites, including the location of Moses's burning bush. At the site of Jesus's tomb, she had three crosses excavated and used a miracle to determine which one was the True Cross. She also discovered the crucifixion nails and placed one in Constantine's helmet and one in his horse's bridle. She is also thought to have found Jesus's Holy Tunic.

Helena explicitly sent her men to locate holy relics. Given what failure would have meant, all these missions were of course successful. Other finds included the rope used to tie Jesus to the cross, more pieces of the cross, and more of the Holy Tunic.

Centuries later, many Crusaders returned with "genuine" relics they had liberated from the holy land. Some were naïve dupes, while others were scoundrels in their own right. The number of nails and pieces of the true cross that were brought back would have been sufficient to build a church or two.

In reality, the chance that anyone would have preserved Jesus's cross or its nails, his crown of thorns, his blood, etc. are extremely close to zero.