Well, there's nothing like planning ahead, is there? Astronomers have predicted, using Newton's theory of gravitation, that the Milky Way will merge (rather than 'smash') with the Andromeda Galaxy in 4 billion years (4,000,000,000 years) but it's never too early to think of names!
The process of merging between the Milky Way and Andromeda will begin a quarter of a billion years before they 'touch' and will continue for another billion years afterward before everything settles down. Due to the extreme distances between individual stars, it is very unlikely that there will be any physical collisions of astronomical bodies due to the merging of galaxies.
Various stars will be relocated, and flung to different parts of the galaxy, including our own Sun. However, it is doubtful any life will remain on Earth to worry about renaming our new neighborhood, due to the rising temperature of the Sun.
Humans may have evolved into different life-forms and left the solar system, and other species may become dominant, or life may become extinct on Earth altogether. Maybe creatures on another world, who have yet to evolve, will name the new galaxy.
Here's a few ideas of my own:
the Andromeda Way
(corporate sponsor's name here)
The All-New Milky Way Galaxy
Here Be Tygers
Milky Way 2.0
New Galaxy on the Block, fo'shizzle
If you can't wait 4 billion years, here's a YouTube film showing a simulation of the Andromeda-Milky Way 'collision'.