chloem sorry, I think you're wrong, and dare I say it, I feel sorry for your Mum's boyfriend. You don't have to like him or have much in common, but small talk and manners cost nothing and make everyone feel better, including your Mother. How can that be a bad thing? Would you treat a work colleague like that? I hope not.
I very much doubt OP is trying to be a Mother to her DP's son. As to why does she care, well I imagine its because being treated badly in her own own is upsetting and frankly, he should be ashamed of himself.
OP, I feel your pain, and sadly, it's a common problem. No, you can't make him like you - and to be honest, he doesn't sound very like-able anyway so I doubt you like him either. Don't try. BUT, you do have to live in the same house for 5 months so the situation can't go on as it is.
I'd sit him and your DP down and say that you'd like to discuss it, as adults. Say to the son that you know that the situation is difficult for him, and you'd like to understand it from his point of view. Tell him that you know you may not like what you hear (and be genuinely prepared NOT to like what you hear), but that you will listen - and try to listen objectively even though it's very hard to do.
THEN, when he's said his bit, tell them both how it's making you feel. Tell him that you are not trying to replace his Mother and that you mean no disrespect to her memory, that you are not trying to diminish the relationship he has with his Father or come between them in any way... BUT you are here to stay, you love his Father and whilst you accept that you and his son may never really get along, that's ok, so long as you all treat one another with basic manners and respect. Anything less is not ok, not acceptable and cannot continue.
Good luck 