I'm not sure this is a case of monopolising the conversation as much as it is being uncomfortable with silence and trying to fill the gaps, so I'm not sure what you can do about that, because comfort on a visit should work both ways. I wouldn't call myself an extrovert, but I'm very uncomfortable with long silences. They feel so terribly awkward, as if we don't have anything to say to each other. Surely the point of a visit is to socialise; if I'd wanted to not talk to anyone, I could stay home alone and do that for free.
For general discussion, I'm curious in situations like this why the introvert or non-conversationalist's preference always seems to take priority. If the extrovert is being rude by talking too much and making the introvert uncomfortable, can't the same also be said the other way round - that it's somewhat rude of the introvert to make the extrovert uncomfortable by not participating more in conversation? Perhaps that's a topic for another thread though.
The OP posted:
She is an extrovert in a room full of introverts, all family. We would like there to be silence occasionally, where someone else could maybe come in and say something. Instead if you want to talk about something you have to jump in when she's taking a breath. And then she'll find some way to relate what you are talking about back to her, and then go on about it, repeating herself again! It's like she's scared of silence or something.
In the situations described by the OP, it sounds like the majority of the people involved believe in waiting until the current speaker has finished before taking the floor, and they are expecting the cue of a pause longer than taking a breath. I.e., they wait for an indication that they will not be interrupting if they start speaking. It sounds like this conversational style works for everyone in the family except SIL. Also, it sounds like others have tried to compromise with SIL's style by occasionally jumping in when SIL takes a breath, otherwise they wouldn't know that she'll relate the comment back into what she was talking about and keep going. If this is true, the OP's situation is not about the reserved style taking precedence over the talkative style. Instead, it's about the minority style needing to adapt to mesh with the majority style.
Also, it has not been my experience that a reserved conversational style "always" takes precedence over a more talkative, outgoing style. In fact, IME our culture has a lot of pressure/emphasis on getting out and meeting people (favors outgoing, talkative style over reserved), networking with strangers at events (again, favors talkative, outgoing style over reserved), attending social events for work teams/clubs/etc. one is associated with (favors extroverts over introverts and outgoing people over reserved ones).
Part of the trouble in dealing with clashing talkative vs. reserved styles is that the talkative style tends to easily overwhelm the other without meaning to. If Person A is uncomfortable with silences and Person B is uncomfortable interrupting before the other person has indicated they're finished, then Person B will never talk, even though both styles are perfectly valid and trying to be polite. This doesn't mean that Person B's preference automatically take precedence, but it does mean Person A must back off before any equal compromise can be reached. Person B cannot create pauses, so their only options are to adopt Person A's style (not a compromise, and leaves B uncomfortable) or stop participating and let A monologue to avoid silence (which defeats the purpose of a conversation). Whereas if Person A is willing to adapt a little by leaving some pauses, then B will have a chance to comfortably step in and hold up their end of the conversation. If A leaves some pauses and B makes an effort not to let the silences drag on too long (even if B would be comfortable with that), then voila! We have compromise and conversation! But B can't make that happen without A's cooperation.
As you said, the point of a visit is to socialize. Neither complete silence nor monologues fulfill that need. Just like you could stay home if you wanted silence, you could stay home and turn on the TV or monologue to the wall if you just wanted a lack of silence. That's why both parties need to adapt.