United Kingdom - Passport & Nationality - British Citizenship by Double Descent (48-5(1)a Mother)
After 1948 and prior to 1983, British Citizenship could be passed two generations (what is referred to as Double Descent) from a UK-born grandfather where the middle generation was born in a place which at the time was a British Protectorate, British Protected State, British Mandated Territory, or UK Trust Territory. It also includes a wide number of foreign countries where the Crown exercised Extra Territorial Jurisdiction (known as ETJ).
This solution requires your mother to have been born between 01.01.1949 and 01.01.1983. They must also have a grandfather born in the UK (or a place that was within the Crown's Dominions, although this will require the applicant to have also acquired the Right of Abode through some other means).
While anti-gender discrimination legislation has been enacted, it still affects the grandparent generation. Accordingly, a claim is unlikey to come from a grandmother. If the relevant parent was the father, then the applicant is British by operation of law (i.e. automatically British), as long as the relevant family members were born legitimately. However, if the relevant parent was the mother, an application has to be made. This route through the mother has become available after recent anti-gender discrimination legislation was passed.
There are two further points to note. Your mother must not have:
suffered from Automatic Loss of Colonial Nationality from the Independence arrangements (though there are some exceptions to this); AND
renounced citizenship of the UK and Colonies.