Is one obligated to pay the value of a damaged frame or the extra time for renting a bicycle, or is one excused for having done so in childhood?

There is no sin upon a child before puberty, based on the Prophet, peace and blessings be upon him, who said: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he reaches puberty, and from the insane until he regains his sanity."

As for liability, scholars have differed on it, but the majority hold that a child is not liable if he was given authority over what he damaged. The Malikis are of the view that a minor is liable for what he damages of another's property, unless the owner willingly gave him authority over it. Al-Adawi mentioned that the non-pubescent and the foolish are liable for what they damage unless they were entrusted with it.

The cautious approach is to pay the owner the compensation for the extra time and the value of the bicycle wheel. If you cannot find him, then give that amount in charity on his behalf.