I would immediately think of the following plants for the trellis:
Pennywort (Hydrocotyl, several species, but any of the mid-sized or larger leaf)
Creeping Jenny (Lysmachia nummularia)
and if the trellis is larger:
Golden Pothos (archway over a door)
Other house plants that can grow with roots in the water, stems in the air. You may have to test some, see what works. Good water circulation is the key.
For a really interesting look grow vegetables this way. Tomatoes can be trellised, and most of the pumpkin/squash/cucumber/melon family. Peas and beans may work, but I am not sure. This would be more of a hydroponic set up, quite large.