Your mother's application was refused under Paragraph V3.6(v) of the rules. This paragraph is a restatement of Paragraph 320 for which subsequent decisions will apply. It's better to research Paragraph 320 because it's the full statement of general grounds.
In addition to the refusal, her application attracted a 10 year ban. This is the standard policy for applications that are deceptive. The fact that you claim to have had an "accident of omission" will not be at all helpful; everybody in the world says something like that when they are caught. There is no recourse of appeal for visit applications of any kind (except Judicial Review in the High Court); and administrative review is for work permits and students (and etc).
They will refuse subsequent applications until the ban expires. Making a fresh application will not work, even if it contains profuse apologies. Everybody in the world apologizes when they get caught and the decision-makers will not be sympathetic. It is a sackable offense to issue a visa when a ban is in effect.
Naturally, you want the ban lifted. It may be possible depending upon the various factors of your mother's case; but it will be hellishly expensive. You will need to instruct a high street and well-established solicitor with a practice area in getting bans lifted (e.g., Sophie Barrett-Brown or Barry O'Leary) and they may refuse the casework if they think it's impossible.
There will be lots of practitioners who will claim to be competent in getting bans lifted, especially in India, but I wouldn't trust them. In some cases in India, they will take your money and then forge a letter from the Home Office saying your appeal was dismissed, and then take your money for an appeal to the Upper Tribunal and then say that was dismissed, and then take your money for a Parliamentary Prayer, etc etc... Remember that you are now a vulnerable person with respect to so-called 'legal practitioners'; it is key critical to keep this in mind when a ban has been issued. This paragraph (about being vulnerable) is the most important part of this answer.
Sometimes people (on the British side) will contact their MP and ask for representation. This strategy invariably fails; the MP will get a letter from the Home Office explaining that rules are rules full stop. But it's free to try if you want.
The other strategy people have used is to marry a Brit and apply under the family rules and ask for compassion. There are very mixed results in this strategy; it can backfire if they decide it is a marriage of convenience.
The last strategy is available if you cannot afford a solicitor, and it is to wait it out. Be aware that when the ban expires, they are likely to continue refusing on credibility and they are entitled to do that. Bans were not put into effect until 2008 so none of them have expired yet and so there is no field experience to rely upon. You can have visits in India and maintain your family relationships that way.
Adding...
Finally, I took a course (yes, there are courses in this stuff, this one was taught by Barry O'Leary) entitled "The general grounds for refusal, re-entry bans and deception" and can provide this extract from the course notes which helps to define the standard applied to the ECO's decision...
The ECG says that 'a false representation is made when an applicant or
third party lies or makes a false statement in an application either
orally or in writing'. It is not enough for the ECO only to 'doubt or
suspect that a document is false'. The ECO must be 'satisfied to a
high standard that a false document has been submilted'. It gives as
examples, a person who says that he is a project manager for a company
earning a significant salary but who is discovered, in fact, to be a
cleaner for the company on a low salary or a person who says he was
never before married but where the ECO receives a marriage certificate
showing that he was already married. Whilst such 'false
representations' might well be material, the ECG, true to the rules
emphasizes that 'the application must be refused even if the false
representation is not material' to the application.
Also Adding...
For a live example of a person being refused for deception and then taken into custody by the Indian police, see UK Border Force on YouTube. You can advance the player to about 10:52 to where the action starts. It skips to Heathrow and returns to India about 15:15. It's a great episode, captured live. The ECO is the fellow in the striped shirt.
