I dont know for sure, but I woudl say that she coudl have been held in a spunging house like Brandon's servant. After all he woudl not be in any better case than Eliza, as regards having soemone to help.. but he was able to stay there for a while until the avenues of payment he suggested were tried out...

Does it say how much she owed? It might not be that much money but in Eliza's unfortunate state, poor, with bad health etc, she might not have been able to pay it immediately and wound up there. She may have been trying to find an ex lover who would help, and if she couldn't get the money paid within a period of time, she'd be transferred to debtor's prison.
I am assuming that that was the case, that it wasn't a huge sum of money but it was beyond Eliza's power to pay it then.. but there was still a hope that soem friend or lover would help.. so she was still there, like B's servant. And luckily for her Brandon turned up and was able to get her out. But yes, children were allowed to stay in debtors' prisons (IIRC John Dickens had his family staying with him when he was in prison for debt) and since a spunging house was often just the bailiff's house, Im sure she'd have been able to keep the child with her.