I believe weeping cherry is grafted at the top of the straight trunk, so be sure that the new growth is from above the graft and not from the root stock -- they should "weep"
Once you are sure nothing is going to grow back from them, start testing to see if the dead looking branches are really dead. They found hollow when you flick them and easily snap with dry break. Be careful because there will be some dead looking branches that actually rant dead and will have live side branches further down.
I go around snapping off the dead twiggy stuff up to healthy branches, then clean up with clean cuts with pruners. I don't like the "bob cut" style a lot of people adopt, so leave mine in a more natural sweeping manner and just keep the branch tips from touching the ground, but I guess "the bob" is more popular.
Yes, it appears the new growth is above the grafted area.
Most of the branches without growth are very brittle, and the twigs I snapped off are hollow in the middle. I like the look of them even though there is no growth, so I'm hesitant to prune off all of the dead branches... which is the majority of the tree.
Is there anything I can do to promote new growth on the other side of the tree? I know very little about this subject, so excuse my newbie questions.
Did something damage the roots on the growing side? Often when a tree dies on one side like that the cause could be root damage on the opposite side.
On looking closer at your pictures I see some damage to the trunk right near the soil level. How bad is that damage? That could be another reason your tree is in trouble.
...or chewed up bark -- rabbits, mice, voles, etc. rodents, deer etc. will strip or chew off bark. If the tree is girdled (Outer and inner fluid transporting layers of the bark completely stripped) the branches with interrupted "supply" would die. This can also happen from weed whackeror lawn mower injury introducing disease or bugs.
Good call butterflylady -- it's not in focus and hard to tell, but there might be visible damage.