Normally we need to see the text of the refusal notice in order to help, but T2 refusals are much more straight-forward and do not have the nuances of visitor applications.
Tier 2 applications that have been refused are eligible for administrative review. The controlling reference point is HC 1025, which was passed into law last year.
Without putting you through the rigmarole of reading HC 1025, here's a summary in layman's terms...
Administrative review has been available to correct case-working
errors in certain Tier 4 (student) decisions since 20 October 2014.
From 2 March 2015 it will be available for decisions on Points-based
System applications where the application (as opposed to the decision)
was made on or after that date. From 6 April 2015 administrative
review will be available for all decisions under the Immigration Rules
where there is no longer a right of appeal.
The Immigration Rules describe the procedure to make a valid
application for administrative review. The Home Office must give
written notice that a decision has been made which is eligible for
administrative review, along with a statement of reasons for the
decision, and information on how to apply for an administrative
An application for administrative review must be made in accordance
with the requirements set out in the Immigration Rules. If it is not,
it will be invalid and will not be considered. Where a person is not
detained, the deadline for making a request for administrative review
is no more than 14 calendar days (seven days for a person in
detention) after receipt of the notice of the eligible decision. A fee
of £80 must be paid. If the outcome of the administrative review is
that the decision on the original application is withdrawn and leave
is granted, the Home Office will refund the fee. The fee will also be
refunded if the administrative review application is rejected as
Source: Immigration Law Practitioners' Association Briefing
Based upon what you wrote, you would need to prove the refusal was a case-working error (or that they failed to provide evidential flexibility) and this may be difficult.
If you decide to go forward, you will need to file for two separate reviews (80 + 80 = 160 total) and you'll need for both to be successful. If one is successful and the other is not, then the person without the visa will have to apply again starting from square 1.
Alternatively, you can file a review for your child and make a fresh application for your wife, or vice-versa. Or make two fresh applications. Nobody, not even a qualified lawyer, can tell you which course of action is better; it's a choice governed entirely by circumstantial practicalities.
For your last question, about how to prove that the bank made an error, you need to show something that the second name is a lawful and recognized alias for you, or you can try to get the bank to change your name and reissue all of the previous statements with the correct spelling. Again, neither of these is better or worse than the other one when it comes to ECO's.