Weekend 24 was shown on BBC Two in the morning at the weekend (originally on just Saturday or Sunday, not sure if eventually both), and started quite soon after News 24 launched (when it was still in N9).
Yes, it began at the beginning of 1998, on Saturdays at 8-9am. I remember someone saying at the tme it had been introduced as people were complaining nobody could see News 24 in those days so it was a chance for terrestrial viewers to see it in a more civilised slot than just overnights. The reason it was only 8-9am was because there was Open University programmes either side of it. It wasn't on Sundays as MOTD, Breakfast with Frost, the OU and kids' shows were all in the way.
It was after 9/11 that it became a regular thing on both Saturday and Sunday mornings, it was initially there just as more rolling news but they just kept it, helped by OU programmes being phased out. As you mention, it was indeed still billed as Weekend 24 on Saturdays - but not Sundays - long after Breakfast had launched. Looking in 2005, though, we then had Breakfast 6-9 and Weekend 24 billed as a separate thing from 9-10.
There were a few other experiments on Saturday morning, in the summer of 2004 they moved the kids shows to BBC2 (eighteen months before they moved there full time) and did a thing on BBC1 at 9am called In The Know with John Inverdale and Louise Minchin which was simulcast on News 24 and interspersed the news with sports previews and features, but it was all a bit of a mess and Invers later said it wasn't very good. And also around 2004, the BBC political review, which also saw the end of On The Record and the launch of The Politics Show and This Week, suggested they should launch a political show aimed at a younger audience so at 9am on Saturday on BBC2 you had Weekend With Rod Liddle And Kate Silverton and The Sharp End with Clive Anderson for a couple of weeks each, but neither caught on and they abandoned that idea.
I always thought it was weird that when The Andrew Marr Show was on its summer break they'd do News 24 Sunday with Peter Sissons, simulcast with News 24, but with exactly the same format as Andrew Marr which wasn't simulcast on News 24, so I don't know why they didn't just do The Andrew Marr Show With Peter Sissons, which is what they do now in the summer of course.
And of course there was that weird semi-Breakfast during the 2000 Olympics, between Breakfast News ending and Breakfast beginning, which ran 7-8.30 on BBC2 and News 24.