Button is the right choice. Not sure why it took them so long to reach that decision... or announce it.
Probably 3 things. Button was due 19 million at this point I don't think they thought they were getting 19 million worth of performance out of him (he is getting less than 10 for '15 now). Button is old, so is Alonso, but Button has been good for a long time he has never been great. Alonso has a lot of skill to lose before he is down to regular performance levels, Button is already there. The third is the rumor that Alonso's pay check might becoming directly from Honda, if that's the case Macca may have thought this would be a nice opportunity to dedicate all resources to the car and not really have to deal with driver salary.
Honorable mention to Magnussen still developing. They still own his rights but if a team comes knocking hard enough they might feel like they have to let him out of his contract. You normally don't want to give up a possible rising star for a driver with only a good year or two left in them.
Not saying they didn't make the right decision. Just saying there is a lot to talk about. Specially if they think Honda will get a jump on everyone else since they know the performance of everyone else's engine which are frozen. Do you really want to pay Button 19 million if you think that Alonso is just going to walk away with the championship?