I don't have any suggestions on your mother, but it very likely could have to do with her past-- many women have internalized the breast as sexual (only), compound that with a shameful/painful event and it makes it even stronger and a harder belief to change.
Maybe by seeing you nursing and growing her (especially since she is the grandmother) perfect grandson, her opinion will shift. I wouldn't push her though-fundamentally it's about her stuff-and really has very little to do with you.
As far as what everyone else thinks--yes, they do think he is old enough for water, juice and solids (and probably diet coke). People have no idea about feeding babies, and whatever they see on tv is what is right. People don't bat an eye when a 3 year old is running around with a bottle, but god(dess) forbid they witness a 3 year old nursing.
Nurse your baby, cite the AAP recommendation for at least a year, then cite the WHO recommendation for 2 years and beyond.