Saying No
The best way to say "no" in any situation is "yes, when."
This advice is a practical application of the general and effective life attitude of focusing on how to find a forward-looking and productive path in every situation. That means mostly setting aside negative feelings from the past, pausing/slowing personal reactions in order to reflect deeply and choose actions mindfully, and then focusing on the problem to be solved—holding people's feelings as of great importance, but always considering them in light of the actions that can make progress, so as not to let feelings simply dictate.
But before you say no, communicate.
Discuss With the Group
Start by having a discussion with all the players, including your sister. Be open and honest about your goals, for example, you might say you're sure everyone wants to have a great experience gaming, and do they mind having a discussion about what that looks like for each person, so they can all get on the same page? How does everyone feel about committing to being present every week? How do they want to handle player absences? Discuss the possibility of working a player's expected regular absences into the story. Don't call your sister out specifically, even though everyone will know who you're talking about. Your goal is to encourage productive, positive expressions from each player to find out how they want to play. Talk in terms of advantages, drawbacks, challenges, and opportunities. Openly list out group goals, and even write them down, perhaps creating a charter of sorts (e.g., having more players and being inclusive are positive, but those need to be balanced with enhancing the overall play experience and creating consistency). Discuss the possibility of playing on a different day of the week. Discuss alternating the time every other week, or playing more than once a week.
In this discussion, try to function first as a mediator or facilitator more than as a person with a personal stake. Practice active listening with statements beginning with, "it sounds/looks/seems like ... ." You could say to one player, "it sounds like you'd prefer people commit to playing every week. I also hear you saying you'd love to accommodate [your sister] being absent every other week, but don't see how that's possible." Or to your sister, "It seems like you feel really disappointed and sad about the possibility of not being able to participate if the group decides it wants players to commit to playing every week." You can include your own perspective after everyone else, but try to do it in the same vein, e.g., "I'd really like to include [your sister], yet I'm worried that the group could shrink to the point where it doesn't exist any more." Labeling positive and negative emotions is a great way to help people understand them, feel them, and then change in response to them—which very often is harder when they remain unlabeled and unspoken.
Overall, the goal is to reframe the issue from "you vs. your sister" to "the will and the good of the group". By openly considering all the options, your sister will see everyone's desire to include her and will start thinking about the group's perspective—they have their own schedule, constraints, desires, and life challenges that need consideration. Once she sees everyone making a serious effort to include her, even if in the end they can't, it is likely she'll feel less upset about it.
Decide, Then Say "Yes, When" if Necessary
Once everyone has had their say, if there is a consensus, the problem is solved. If the group is okay including her in some way, do that. But if consensus can't be reached, then someone has to make a decision.
If most feel that including a player who can only attend every other week would be disruptive to how they want to play, then here's how you can say "no" by saying "yes, when" to your sister (and this plays well for your mom, as well):
"The group would love to include you as soon as there's a weekly time that works for the majority. Let's revisit this once a month to see if things change or if another time has opened up so that you can join."
Then, Stay Firm
Don't waver. Repeat this as needed over and over like you've gone brain-dead, but with appropriate and honest feeling: "The group would love to include [sister] as soon as there's a weekly time that works for the majority." Fully accept, in advance, all consequences that come from holding this position. You'll find that the peace you feel with such pre-acceptance enables you to speak with great compassion and a quiet, firm resolve that will be surprising to your family, and they'll almost certainly react differently. The response to all argument, challenges, criticism, and complaints is the same phrase.
(That's called "assertiveness," which is not a fighting attitude but a mindful deliberateness that calmly and masterfully chooses actions, then accepts consequences for them in advance.)
