You need to get some info about the specific model # of your mother's (old) phone.
Not just the name of the phone, like "iPhone 6S", but the actual model number. This specific detail will be available in the settings menu of the phone.
It's important because the 'same' phone can be manufactured so that it uses different bands (think radio frequencies) depending on the country that it is sold in. For example, iphones sold in the US don't work particularly well in Japan, since the range of common bands--those used in both countries--is not that wide. Believe it or not, some phone like an iPhone 6s that is manufactured for the US market, and an iPhone 6s for the Japanese market, are not the exact same phone on the inside (same situation for other makers).
In addition to the country where it was made to be used (and then the country in which you want to use it), different carriers here in Japan also use different bands (different frequencies). So your mother's old phone may work well (or better) on the docomo network than on softbank or AU. Or vice versa.
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Once you have the specific phone model, go to a site like
WillMyPhoneWork.net - Check if your phone works on a network and plug in make, model and number, and then go to the target country and carrier/network to see if it will work. (how well it might work)
While you could go with a major carrier, if that's what the results say, you should also consider an MVNO, and getting a 格安SIM. Depending on your choice there, the monthly plan cost could be some fraction of the cost of a major carrier. For example, for a number (voice) and 3GB/month, I pay ¥1400/month to IIJmio (which uses the docomo network). If you're a heavier user, you can pay more for 5, 10, 20GB/month as you please.