There are no charges for opening a minor account at Zerodha.
Buying securities
You cannot buy stocks directly in a minor account. The guardian can transfer existing securities from their demat account or buy and transfer securities into the minor's account using CDSL easiest or Zerodha's gift feature. However, gifting charges will apply.
Selling holdings
You can sell holdings from the minor's demat account with no brokerage charges. However, a minimum of ₹0.01 will be charged per contract note. As per taxation rules, a service can not be provided for an absolute 0 fee, due to which a token amount of 1 paisa is charged.
Depository Participant (DP) charge is applicable whenever shares are sold from the demat account.
To learn all the charges involved, see List of all charges and taxes.