Perhaps you could argue on the material facts that they got wrong, that is, the fact that your child don't have a status in the UK.
Secondly, you could agure that Rule 248, which provides for access to a UK child, which is closely linked to Article 8 is breached, by this refusal. Essentially, this would mean, you British child could potentially face deportation, wrongly. This would georpodise the family life you already have with him, as you will be unable to afford travel abroad, and this is impractical.
Thirdly, you could also, if applicable to you, argue that your wife contributes to you wellbeing, and as you are suffering from ill health, her presence, together with that of your child is essential to you wellbeing. That is, if that is the case of course.
I suspect rule 248 will be a more stronger case, although not a clear cut one. If your child is taken abroad, as your wife would be the prefered person for him/her to be with, it would essentially mean that contacts with him is curtailed, which is in breach of rule 248.
http://www.ukba.homeoffice.gov.uk/sitec ... iew=Binary
http://www.ukba.homeoffice.gov.uk/sitec ... iew=Binary
Just an idea. Have a look at this and see if it applies to your case. The fact you are unemployed makes your position a bit precarious.
All the best with the appeal.
Chikwamba is a good one, but it would have been stronger, if your child was in school , and asking mum to go abroad would unsettle him. You can nevertheless use it and see how it goes.
There are free legal service in the UK, like Community Legal Service, or the Citizens Advise Bureau. They can be pretty helpful. You should try and give them a go.