I would just look at the cheer bonuses:
http://ffxiclopedia.wikia.com/wiki/Monster_Rearing
I'm not really interested in collecting items (and you will get junk a lot if you do, but also randomly nice stuff).
The skill gain bonus are ok (better for crafting than battle skills).
Alter ego bonuses are meh.
Combat bonuses are also underwhelming.
the CP bonuses are the best IMO, still trying to get Jumbotender for movement speed.
In anycase cheer bonuses are free, (well you need to feed it, but its cheap (usually there's a few expensive evolution foods) and only takes a couple min. per day max). You can only use 1 at a time, and they costs shining stars to switch.
So whatever suites your fancy but the 3 reasons would be: item collection, cheer bonuses or just the aesthetics of having x monster lumbering around your garden.