This is not great news. They got her on V 4.2 and V 4.3, which means they challenged your credibility as a sponsor. We don't see these grounds very often, but they arise when...
- the applicant has no chance of qualifying on their own; AND
- they don't believe the sponsor really has the capacity for
For the first item, you can get some tangentially related background from Should I submit bank statements when applying for a UK Visa? What do they say about me?.
They also refer to a prior refusal in 2006; that's 11 years ago which means it must have uncovered something significant for them bring it up after such a long time. It suggests that your mother's circumstances have not actually changed and the possibility of a lingering credibility issue. They like to point out that successive applications are examined under the assumption of fresh circumstances, but if the circumstances have not changed or if the applicant has established that they are not a bona fide visitor, they will keep bringing it up.
Jamaican applications are decided at the British Consulate General in NYC, and you would already be aware that Jamaican applicants face additional difficulties owing to long-standing performance issues. OK, that's not your mother's fault, but it's part of WHAT IS, and should be accepted at face value.
Circumstances have now changed and I and my wife will be joint
Adding sponsors does not change the circumstances. You can read UK visa refusal on V 4.2 a + c (and sometimes 'e') for a somewhat related discussion about sponsorship and circumstances.
Even though I'll be paying for her travel and accommodation and
looking after her needs for the two weeks, will they refused her on
her own circumstances?
Yes, it is the applicant's circumstances that are paramount in assessing an application. This remains true regardless of the sponsor's capacity; it's a common misconception that an applicant is relieved of their onus because of adequate sponsorship and we have literally dozens of refusals in the archives here where ECO's have pointed out that the applicant is never relieved of their onus.
She works full time. We're joint owners on the property where she
lives. I send her money every month to pay the mortgage, bills etc. My
sister lives with her and she's a adult. She has sisters, brothers,
nieces and nephew that live in the country.
These things are great for establishing that you and your mother have a durable and tenable connection, especially your remittances. These should be offered as primary evidence if your mother wants to apply again (which I would NOT endorse by the way).
I'm thinking of applying again and was wondering what are her chances
We don't take bets.
Your best option at this point is to instruct a UK based solicitor with an established practice area in family visit refusals (especially Jamaican nationals) to represent the next application. You can use the search engine at ILPA (the trade association and a registered charity) to help get a referral. But take note that the bulk of them are fee-earners and there is no legal aide support for visitor applications (and we're talking some serious coinage here).