It's only going to be the old Java (5) on your (ME) machine that would be a problem and anything past JRE 6 on your mom's machine will (should) be fine.
Java 6 has just updated to update 4 so that's probably what you've seen trying to automatically update on your mom's machine. (and that's Vista doing it's job)
It's not vital she has the very latest update immediately. But if you tell her to go into the Vista control panel and look for the Java icon and bring up the mini Java control panel, and on the first tab (General) click the About button and it will tell you what version you have (same goes for you in ME).
Go further down and you'll see Temporary Internet Files and the Settings button, click that, and click the Delete button, click ok.....this deletes the Java files stored for the browser, it's possible one of those is corrupt. If you have the same option in ME you could try doing the same.
Next, click the Update tab and click the Update Now button, that will show you what version you have too, and will also update to the next available update manually, when it's ready.....so if it doesn't update one day, or the next - go back a week later and it will probably update.
Like I said before, current version of Java for your mom (and everyone else on XP/Vista) is Java JRE 6 Update 4.
It's also worth you both downloading and installing CCleaner and clearing out the entire browser cache, cookies, etc (it does both Internet Explorer and Firefox)
There is also an FAQ on the jigzone site: jigzone FAQ which touches on all this.
