I'm going to assume your mom's set only has an antenna input, and does not have RCA jacks on the back (an AV input.) We'll also assume that there is no VCR or DVD player involved (which adds another layer of options and complexity.)
So, given those assumptions, as an earlier poster stated, you really should have a converter box with analog pass through - some boxes have it, many do not. It should say in the manual that came with it (although I'm sure some boxes come in inadequate documentation.) Here is why you want analog pass through: The dish box sends an analog signal (probably on channel 3 or 4) to the old analog tuner on the TV. So if the digital converter has analog pass through, you connect the antenna coax from the dish box to the input on a splitter/combiner, and you connect the coax from the antenna to the second input on the splitter / combiner. Then you connect the output from the splitter / combiner to the input on the digital converter, and you connect the output from the digital converter to the TV. Now, when the digital converter is turned on, you can tune in your over-the-air channels, and when it is turned off, it "passes through" the signal from the dish network box, and you can watch the dish. If the box does not have digital pass through, you can get an a/b switch at radio shack that would allow you to switch from the dish box to the digital box, or you might try using a splitter/combiner to combine the signal from the dish box with the signal from the digital converter. This might work if one is set to channel 3 and the other is set to channel 4, but I'm not sure - there might be too much adjacent channel interference.