If you can think of a word for all three of them, I'm interested.
Well, how about 'God' as the word for the 'Three combined' and 'manifestation' (of God) for each, individually?
Let me explain my thinking: The fullness of the nature and form of God is utterly beyond human understanding and comprehension - and those who try to explain, justify, or criticise God's actions (or lack of action) in human terms are paddling their canoes uphill (so to speak).
In the Hebrew Bible, various theophanies occur, in which God 'manifests' himself in various ways - usually involving thunderstorms, earthquakes, volcanic eruptions, or other powerful works of 'nature'. But there are also other, more intimate theophanies: the Burning Bush, from which he reveals himself to Moses; the pillars of cloud/fire, by which he identifies himself as Israel's guide and protector, to name but two. But the overarching doctrine of the Hebrew Bible is that "no man can see God" (and survive the experience).
With the arrival of Jesus, we have a fuller revelation of God and the way he interacts with his Creation: three different manifestations of the same, single God:
The "Father" is (in some ways) equivalent to the Hebrew YHWH - the invisible, indefinable King of Heaven.
The "Holy Spirit" (a term already used numerous time in the Hebrew Scriptures) is the force by which God can (and does) inspire humans to proclaim his word, perform miracles in his name,
et cetera. In the NT, it is also used (sometimes) as the force behind theophanies (like shaking houses where the apostles were meeting).
The "Son" (Jesus Christ) is God's manifestation of himself 'among us' (Immanuel = "God with us"). It is interesting to note that Jesus often defers to 'the Father' and (on at least one occasion) elevates 'the Spirit' to a 'holier level' than himself (those who sin against the Spirit cannot be forgiven).
Of course, there is nothing to prevent God from manifesting himself in any number of other ways - but these Three are sufficient from a doctrinal perspective, and they are the 'canonical' manifestations that he has given to us and proclaimed to us.
Why Three? Well, that's not for me to say - but there are numerous passages in Hebrew Scripture that allude to the Trinity: The three 'men' that Abraham saw when the LORD appeared to him at Mamre (Gen 18); the "Holy, Holy, Holy" song of the cherubim in Isaiah's vision, and others.