I'd get the .44, because I already have a .22 I like.
What is she wanting to do with the gun? Targets shoot, shoot squirrels and bunnies, shoot deer, or all the above?
Handloaded, the .44 isn't outrageously expensive to shoot, and you can make ammo suitable for all the above. I know this because last night my wife informed me that as we get back on our feet, she wants a .44 Magnum rifle to replace her .32-20 we just had to sell.
Everyone knows that the .44 is TWICE as powerful as the .22....
I still voted for the 10/22, though.
And if you have enough money for both...then use the money you would spend on the handi rifle to buy some upgrades for the Ruger. A nice scope and some internal bits.
You'd still have some left over for ammo.
Sure, plinking with a .44mag rifle would be fun... But those Handi Rifles aren't for the recoil shy, that's for sure. Plus, there are only a few animals here in the state that are "hunting eligible" with the .44...provided you're hunting for the meat and not just to kill something.
The 10/22 is the right choice for almost everyone. Ammo's way cheaper -- great for practice and plinking.
I have a .44 Handi-Rifle and it's great for deer, but I don't plink with it that much. I don't think it kicks too bad (takes a couple dozen rounds before I feel it at all), but if your wife is at all recoil shy the 10/22 will be a lot more fun.