I have heard this phrase in Italian blogs, Instagram accounts etc.., I know a little bit of Italian and couldn't co-relate this to anything. My hunch is that it means stay beautiful (or well). Does the baell come from Neapolitan or some other dialect? In what circumstance is this phrase used?
It seems to be a new entry in the language of youths. According to http://www.blitzquotidiano.it/foto-notizie/giosada-vince-x-factor-foto-ma-che-vuol-dire-baell-2339679/ this word has been invented by an Italian singer/actor named Giosada as a variant of “bell” (Bari dialect for bello).
The word spread rapidly in the social networks (never heard before, to be honest). So stay baell could mean be nice or variants thereof.