"~도 그렇고" => ~도(as well) 그러하고(being so) => (something) being so as well.
It is used to give a reason or background for what follows (sometimes omitted), not in an obvious and direct way but in addition to other existing ones not mentioned. Often it suggests one has other things on his mind about the subject.
So 그것도 그렇고 is like "that being so (among others)", and 이름도 그렇고 "what with the name being like that".
- 걔는 몸이 아파서 그 일은 못 시켜.
=> We can't have him do that work as he is unwell. (gives a clear and direct reason)
- 걔는 몸이 아픈 것도 그렇고 그 일은 못 시켜.
=> What with his health and all, we can't have him do that work.
=> Given his health among others, we can't have him do that work.
- 경제도 그렇고.
=> The economy (being) in that way (so...).