Gamera
... I'm not sure if you could paint the USS Eldridge some odd colour, maybe with glow in the dark paint or do something smaller scale and cast it out of translucent plastic or just have some weird lighting effect with LEDs in a shadowbox. ...
Could use holography. There are some modern doable methods. For example
http://www.litiholo.com/hologram_kits.html
So one minute it is there, and then it is not.
You could add other bits in background, like you mentioned with LEDs in a shadow box, and could include rotating prism, with LED lasar pointed at it, flashing beams of light through a thin film of smoke...that was my idea for the Arc of the Covenant.