Yes, it's weird John's mother is so completely absent from canon. Probably because of lazy male writers obsessed with their own daddy issues! It's hard to see her as at all part of John's life when you think about the siege video. John didn't want to send a message to anyone.

I like your ideas and esteefee's. Other things I've seen in fanfics are John's mother as an alcoholic or using pills to cope, often as the marriage to Patrick was a disaster.
There's potential for hurt/comfort fic if she wobbled into the funeral, drunk, and then made a scene. Dave would be swearing under his breath because her minder/paid companion was supposed to keep her out of sight of the guests. Maybe John tries to get her to come away even though it hurts him to be around her when she's like that. Maybe she says something hurtful to him. Eventually, Rodney gets John to leave, and takes him back to their hotel where comfort ensues.

Or...she's also a mover and shaker like Patrick, and they divorced years ago when John was a kid. Patrick made her give him custody and avoid the boys as part of the divorce settlement, which was in the millions, and which she used to set up a rival company. She turns up at Patrick's funeral and announces that she's bought Sheppard Industries and Dave now works for her. She and John quarrel about her abandoning them and putting her career first. Rodney intervenes with some well-chosen words, and later comforts John.

Or there are the fics where she's actually an Ancient!