But if you still feel it would be better to take action on your own, here's what you can expect.
A creditor does not have to accept any payment arrangement other than the one agreed to in the original credit contract. With that said, your son's creditor has limited alternatives for collecting the money owed. Your son is unable to pay the bill, and he currently has no income to garnish. Because of that, you may be able to make arrangements to pay what you can afford each month to satisfy the bill.
Contact your son's creditor, and ask to speak to a supervisor or someone who is authorized to make a repayment plan agreement. Explain your son's situation and that you would like to pay what he owes. State what you can afford to pay each month, and stick to that amount. Be aware that the creditor may attempt to get you to agree to an amount that is beyond what you can afford. Let them know you have reviewed your monthly budget, and you cannot pay more than the $40 to $60 per month you have. If you are able to come to a repayment agreement with the creditor, be sure they send you the agreement in writing. Make sure the agreement explains how they intend to report the account on his credit report.
Depending on how late the account is already, they may be able to waive the delinquency and report the account as "paid as agreed." Also, be sure you aren't being made legally responsible for the debt. I wouldn't want you to suffer if, for any reason, you can't make the payments. You want to pay his debt, not make his debt yours. Once you have received the written agreement, you can begin making monthly payments.
Should you be unable to come to an agreement with the creditor, thank them for their time, and end the conversation. You have the option to set the money aside that you can afford every two weeks until you have saved enough to pay the $300 debt in full. At that point, you could contact the creditor again and negotiate payment in full on the account. Again, get the full payoff agreement in writing before you send in the payment.
You are a good mom to want to protect your son's credit. But before you try to do too much, please think this through. Ask yourself if it's better to let him resolve this small problem once he gets out.