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Making good use of stopovers (when permitted) can really stretch one's miles. For example, if the OP's mom one wanted to fly KIN-MIA-NYC-YYZ on AA, with stopovers at MIA and NYC, AA would charge at least 40,000 miles for a Coach redemption, since AA does not permit any stopovers on such a routing. However, the very same AA flights, if booked with BA miles, would cost only 17,500 miles, as BA permits multiple en-route stopovers -- even on partner awards.

On the other hand, AA can be very good for redemptions between Europe and the U.S., as AA does permit a single stopover at the North American "gateway" on such an award. For example, Europe-JFK can be had for as few as 20,000 AA miles; but for the same miles, one can get Europe-JFK (stopover) - Hawaii/California/Alaska, etc.

Stopovers do not have to be for just a few days. The OP's mom, for example, could "stop over" in NYC on her way back from Jamaica for six months, before continuing on to Canada.

In short, it is important to study the details of each program to determine which one offers the best value for any particular award itinerary.