salix

IT may "live" in a barrell but it won't be happy for long and will soon look distressed

A Salix is a willow and they can easily be increased by propagating. Just cut off about 10 inches of a new stem and plant in a pot of compost, keep damp and it will soon root. I'd do several around the edge of a large pot.
You could try and move it, but I doubt it would be happy and survive. It would best be done in winter when the tree is dormant.

If it has been in the ground for eight years the rootball will be pretty large. Moving it into a planter could be a problem. If you have to move it, it would probably be happier in the ground.

Salix needs a lot of water, that's why they seem to thrive best beside streams and ponds.
They grow very fast as well, so need root space. I agree with others, a pot isn't such a good idea.
If you want to use a planter try one of the miniature acers, or small evergreens.

The salix was in memory of their mum, another plant won't do. I would suggest trying cuttings as suggested by J52, and if that does not work perhaps plant a new plant of the same variety in memory of mum - not quite the same as keeping the original but better than trying to move it and watching it die.

Willows do really sprout easily from 'sticks'. I bought a willow fish sculpture, on a rod, put it over the pond and eventually the wind blew the fish away.
The following spring a pretty little willow was growing by the pond. It was the rod bit that had rooted!

As J52 says. They have planted a living willow structure at GS's school, just by putting bare willow rods into the ground. They are growing nicely. Similarly, there is a living willow fence at the playground in our local park.

I gathered up some willow clippings when a local willow structure was being pruned (with pruners permission) stuck them in the ground at home and they've all grown.

Willows do indeed sprout from almost anything. I remember seeing a willow gate post that started sprouting once!

...pruner's..

I haven't been recently but there were some Willow statues related to fishing in a seafront garden at Rhos-on-sea.