It is a mystery ... like REALLY.
The word "person" caries too much "independence" (as if the Father-Son_Spirit were not 100% united in purpose and fully ONE GOD), yet most alternatives that express the "unity" will then do injury to the "independence" that does exist (the Father is not the Son who is not the Spirit, they are 3 separate "actors" as we see in the Baptism of Jesus when the Son stands in the water, the Spirit descends through the air and the Father speaks from heaven all at the same time).
So 1 being (GOD) with 3 minds (acting in agreement). That is why the Athanasian Creed goes to such elaborate lengths to describe what it IS and what it IS NOT. This understanding is not what PEOPLE wanted or figured out ... it is just an admission of what Scripture reveals about God (Father, Son and Spirit).
Athanasian Creed
That we worship one God in trinity and the trinity in unity,
neither blending their persons
nor dividing their essence.
For the person of the Father is a distinct person,
the person of the Son is another,
and that of the Holy Spirit still another.
But the divinity of the Father, Son, and Holy Spirit is one,
their glory equal, their majesty coeternal.
What quality the Father has, the Son has, and the Holy Spirit has.
The Father is uncreated,
the Son is uncreated,
the Holy Spirit is uncreated.
The Father is immeasurable,
the Son is immeasurable,
the Holy Spirit is immeasurable.
The Father is eternal,
the Son is eternal,
the Holy Spirit is eternal.
And yet there are not three eternal beings;
there is but one eternal being.
So too there are not three uncreated or immeasurable beings;
there is but one uncreated and immeasurable being.
Similarly, the Father is almighty,
the Son is almighty,
the Holy Spirit is almighty.
Yet there are not three almighty beings;
there is but one almighty being.
Thus the Father is God,
the Son is God,
the Holy Spirit is God.
Yet there are not three gods;
there is but one God.
Thus the Father is Lord,
the Son is Lord,
the Holy Spirit is Lord.
Yet there are not three lords;
there is but one Lord.
Just as Christian truth compels us
to confess each person individually
as both God and Lord,
so [universal] religion forbids us
to say that there are three gods or lords.
The Father was neither made nor created nor begotten from anyone.
The Son was neither made nor created;
he was begotten from the Father alone.
The Holy Spirit was neither made nor created nor begotten;
he proceeds from the Father and the Son.
Accordingly there is one Father, not three fathers;
there is one Son, not three sons;
there is one Holy Spirit, not three holy spirits.
Nothing in this trinity is before or after,
nothing is greater or smaller;
in their entirety the three persons
are coeternal and coequal with each other.
So in everything, as was said earlier,
we must worship their trinity in their unity
and their unity in their trinity.