Are you currently housing them in a test tube? Make plans on needing a second formicarium in a month or so, or you can simply put the test tube into an outworld and add several additional test tubes.
Like this:
https://encrypted-tb...f96OSVJDDGjXlZN
http://6legs2many.fi...haenogaster.jpg
I am working on another formicarium design that works well with smaller ants that multiply fast. It is very easy to make. I will post a how-to-build thread once I know it works well (My Tapinoma sessile seem impressed with it).
As for name suggestions, what about picking names that start with "L" for Linepithema. Such as Luna, Laila, Lacey, Lilian, or Lassie.
I don't really give my queens actual names, they tend to get "descriptive names" such as "big red", "flighty fusca", or a 2-3 word description so I can keep them straight. Although, how you are going to tell the 3 apart is beyond me, unless you mark them like the lab scientists do.