Unless you plan on truly skewing the "life expectancy" averages, I don't think it really matters what we think. But still, hypotheticals can be fun, and this certainly does qualify.
It is very unlikely that the resulting galaxy will resemble in any way, either of the originals. So, incorporating one or boh of their names, would only hold really hold "historical" or "nostalgic" purposes.
Then again, Andromeda and Milky Way don't really describe either of them as spiral galaxies to begin with, so my previous paragraph is up for scruitiny.
"Andromeda" might be a little more difficult to hold on to. It would refrence an ancinet constellation that (a) would no longer exist, and (b) would be irrelevant if it was a part of your own galaxy, seeing how it got it's name based onthe constellation it was observed in back in the days when humans on earth were naming objects in the sky.
If you were going to do a name merger, my personal prefrence would be "The Andromeda Way."
If the universe is the limit, I'm going to have to do some brain stretching exercises before I attempt this one.