Would it be possible to start a garden near the top of Mt. Everest?
Certainly one could start a garden anywhere but being successful at growing something I think is the main point of your friend's question.
The closer you get to the top the more extreme the weather. If you wanted to start a moss garden there is a moss that is found at 21,000 ft or so. At about 20,000 ft there is a non-native alpine cushion plant you might grow but you have to go down to 18,000 ft for a native alpine cushion plant. (At least these are all below the death zone of 26,00 ft so you could tend the garden.)
You might want to be rather practical about all this and do your gardening in Kathmandu (4,600 ft) - close to Mt. Everest, longer growing season (by comparison) and better access to goods and services. Here you would be able to have vegetables as well as ornamentals.
You can start a garden at 249 miles above the earth as has been done several times in the International Space Station. Both vegetables and zinnias have been grown there. So, if you have such an environmentally contained structure near the top, yes, you can start a garden near the top of Mt. Everest.