"How about a nice game of chess?"

I thought the computer went by Joshua, since that's what Dr, Falkan referred to it in the last scene of the movie:

https://www.youtube.com/watch?v=s93KC4AGKnY

I don't think we can conclude any name.

Falken saw the computer as a surrogate for his dead son. In that scene he said "Hello, Joshua" but the screen only shows "Hello". Lightman and Mack never knew the name WOPR, at least not before the military got involved, so the only name they had to go on was the password. I don't think the computer ever mentioned a name for itself.

“Freedom to keep playing is a Bombadil-like equilibrium of worthlessness: the un-valued here-and-now.” — RF