The only use the gun came of was to destroy the machine and send the Time Lords back in the time war. But couldn't the sonic screwdriver have done the same?
Maybe she thought he'd need it as a weapon.
Perhaps she foresaw the situation he'd be in - caught between Rassilon and the Master, each one urging him to shoot the other - but didn't realise until the moment itself that he could shoot the machine instead. (I'm using "foresee" here in the mundane sense that we can always foresee certain events if we're smart enough, not in the time-travel sense.) She wanted him to have the gun in case things got hairy and he really did need to use it to shoot someone. It could easily have all happened differently, say if Rassilon or the Master had been faster to attack him before he could destroy the machine.
Maybe it's another bootstrap paradox.
If there was some timey-wimey stuff involved and she'd actually seen the future - or indeed if her appearance on the television to Wilf was after (in her timeline) the big showdown - then she would know for certain that the Doctor would show up armed with a gun. She told Wilf to arm him just to make sure that what was already going to happen was really going to happen. It doesn't seem to make much sense this way, but hey, bootstrap paradoxes don't. Wibbly wobbly, timey wimey.