Do they have a ship name?

Well, they don't really. Although, a lot of people would call them BB, B/B o B&B. I dunno if you'd count them as ship names?

They're very rarely referred to as anything other than B&B, B/B, BB, etc., but I have seen people use Semperance a few times. :)

hmm I like BB and I like Semperance. But ive gotten used to saying Brennath :)

Well usually people call them Booth and Bones,but for short they use BB. If you're a fan of fanfics then definitely look for them as "Semperance - Seeley+Temperance" .

i've heard people use Brooth. easier and shorter than Semperance. in my opinion. i like Brooth. i use it... :D