Son has to execute registered gift deed or relinquishment deed for his 50 per cent share in flat 

Mere affidavit on stamp paper is not sufficient to transfer ownership of flat to mother 

Son should execute gift deed or relinquishment deed 

For society records affidavit by son relinquishing his share in flat should suffice 

The deceased was the owner of the old flat and holder of share certificate

Upon his demise intestate the flat devolved on his legal heirs being his widow and son

Before the old flat could be formally transferred to the names of the legal heirs, the old flat came to be demolished as the society went for redevelopment 

Since the widow is one of the heirs of the deceased,  the society transferred the share certificate in her name 

Thus she was holding the share certificate and rights to be allotted the new flat in lieu of the old flat as a trustee or in trust for the son who is the 50% owner 

Thus even though the paaa is executed in favor of the widow , she is merely holding the new flat as a trustee 

For proper conferment of title the LOA holder who is the son is required to execute and register transfer deed in favor of the heirs of the deceased for transferring the assets of the deceased to his heirs 

That formal transfer is not yet done 

So the LOA holder son in his capacity of an administrator has to formally transfer the old flat and shares thereto and the right to be allotted the new flat in favor of the widow , with the son in his individual capacity joining such transfer deed as a confirming party 

So the LOA holder will be the transferor in his capacity as an administrator of the estate, the son in his individual capacity will be the confirming party and the widow will be the transferee. Thus with the consent of the son (in his individual capacity) the son in his capacity of an administrator will transfer the old flat and shares and since the old flat is already demolished , the right of allotment in respect of the new flat , will also be transferred. However since the paaa is already registered in favor of the mother,  this transfer deed will merely complete her title to the new flat and the society can then admit her as an exclusive shareholder in respect of the new flat 

Index 2 only gives details of the parties and description of property. Index 2 does not make the mother owner of the new flat. In any event the paaa was registered in her favor as a provisional member and trustee of other heirs (in this case the other heir is the son). 

The stamp duty on the transfer deed is nominal Rs. 500 plus registration fee. 

Release deed can also be made. There is no bar against it. The son is entitled to 50% estate of the deceased. So he can very well release his 50% share in favor of his mother. The registrar's refusal to register such release deed is outright illegal. He has no such statutory powers. He has no business to look into the title of the releasor son. 

The son is one of the legal heirs to his deceased father  hence as you rightly observed  he is entitled for 50% share in the property.

If he decides to give away his share in the property to his mother, then he can execute a registered release deed relinquishing his rights in the property.

A mere affidavit or a NOC is not sufficient to relinquish his rights in the property.

In the release deed he can mention that no consideration has been received. 

The proper procedure to be adopted in this circumstance is that the son should relinquish his rights in the property by executing a registered release deed or he can transfer his share in the property to his mother by executing a registered gift deed  and both can be without any any consideration.

If the flat registration has been made on her name itself directly after redevelopment  and if the index - II is also on her name, then she becomes an absolute owner of the entire property.

In that situation, in order to become the primary member of the society, she can obtain NOC from her son by an affidavit duly notarized, which should suffice the requirement just because of the reason that he is one of the legal heirs of the deceased owner.  

a letter of administration grants the appointed administrator the same authority as an executor; allowing them to access, manage and distribute the deceased's assets, debts, and other financial matters.

section 317 of succession act 

An executor or administrator shall, within six months from the grant of probate or letters of administration, or within such further time as the Court which granted the probate or letters may appoint, exhibit in that Court an inventory containing a full and true estimate of all the property in possession, and all the credits, and also all the debts owing by any person to which the executor or administrator is entitled in that character; and shall in like manner, within one year from the grant or within such further time as the said Court may appoint, exhibit an account of the estate, showing the assets which have come to his hands and the manner in which they have been applied or disposed of.

(2)The High Court may prescribe the form in which an inventory or account under this section is to be exhibited.

From your latest post it can be understood that you have properly understood the suggestions made. 

However,  please note that there was no necessity to obtain LOA if there was no more than one property left behind by the deceased and that there are only two legal heirs surviving the deceased. 

As this is the only property,  and since there's no financial transaction involved in this property,  for a formality sake you may file a memo before court expressing NIL account and can file a memo to withdraw the LOA as infructuous owing to settlement between the legal heirs. 

The society should be explained that there's no necessity to execute any release deed since the registered deed is on your mother's name. 

Just submit a NOC in an affidavit. 

In the said flat the mother wife and child will have equal rights as per Hindu succession law

Releasing 50% rights by way of an affidavit would not be proper to confer proper title on the widow 

Either a duly registered release deed or a transfer deed as explained above will confer proper title on the widow 

After the grant of LOA the administrator has to file a report stating how he dealt with the assets of the deceased ...this procedure hardly anybody does after being granted LA.. so don't bother much ..if and when you get any notice from the court,  that time you can file a simple report explaining how you dealt with the various assets of the deceased 

- No, an affidavit for affirmation is not enough for transferring/releasing the share 

- If the registrar is creating trouble then you can take a letter from the society to show that she is only a provision member and not owner of the property in the record of society , and hence the registration of release deed is necessary.  

- If society is ready to accept the NOC /affidavit then you can execute the same