What names are there?
Microsoft calls this a List Builder. As does the 2006 book 'Designing Interfaces' by O'Reilly.
Google calls this in Material Design UI a "transfer list" (or "shuttle").
OpenFaces (an AJAX based component library) calls this TwoListSelection.
Virtuosoft has a couple variations on what they call Dual Listbox. Codeplex also has a version called Picklist (Dual Listbox). Geodan names their version pretty similar; it's called Dual List Box on Github.
Someone else named their implementation Listswap.
There are a bunch more online that have less catchy names such as Moving Values Between Select Boxes, Moving Items Between List Boxes and Item Swapping from ListBoxes. These aren't object names, but article titles.
Which is most used?
Microsoft and O'Reilly are a pretty well known companies so they seem a good starting point. But while googling the term "list builder" gives a whopping ~475.000 results, many are not what we are looking for. Many link to email/marketing tools such as signup sheets for news letters (grow your client list) or adding content (a list of concerts for a band).
"TwoListSelection" gets ~2000 hits without a lot of irrelevant links. But, many of these link (in)directly back to OpenFaces, so it's mostly based on just them.
Googling the terms "Dual List Box" and "Double List Box" gives ~3000 and ~6000 hits, which isn't a whole lot... but most (all?) of those actually link to appropriate content. Different implementations of this GUI element in HTML, Perl, VB2008 and so on.
Listswap gets about ~29.000 hits, but again many false positives. The majority of these are not about swapping items between two lists, but swapping places of items within a single list. "Apple, Bacon, Cheddar." becomes "Apple, Cheddar, Bacon."; not quite what we want.
My Pick: Dual Listbox
Though it might not have the most hits, it has the 'most' hits without having to sort through a lot of unwanted results. It's also got some traction in different languages, as well as being used by several sources. This makes it a pretty healthy choice; not too dependent on a single organization and easy enough to find.