In the statement, what does he mean by the phrase "to divine a purpose"?
To 'divine' a purpose is to act as a god: it is to tell a story that shapes reality according to one's will. From George Herbert's A Dialogue-Anthem:
Chr. ALAS, poor Death ! where is thy glory ?
Where is thy famous force, thy ancient sting ?
Dea. Alas, poor mortal, void of story !
Go spell and read how I have killed thy King.
Chr. Poor Death ! and who was hurt thereby ?
Thy curse being laid on Him makes thee accurst.
Dea. Let losers talk, yet thou shalt die ;
These arms shall crush thee.
Chr. Spare not, do thy worst.
I shall be one day better than before ;
Thou so much worse, that thou shalt be no more.
We can see this as Death claiming the final right of telling a story (Star Trek TNG: All Good Things...):
All good things must come to an end.
The Christian retorts:
Your power was lost by attempting to kill God, opening the way for mortals to become gods.
Suppose, however, that the Christian is wrong. Then the power we do have is to tell a story that will last until death. In The Denial of Death, Ernest Becker describes people who attempt a "sui generis project", which will allow the person to somehow "live forever". You can think of this as a "legacy which lasts forever". What happens if we take a step down from this "sui generis project"? I suggest that is approximately where Einstein lies:
man is here for the sake of other men - above all for those upon whose smiles and well-being our own happiness depends.
This is a pretty standard enlightened self-interest stance. It says that I will live for a finite period of time, and therefore there are ways to optimize my experience in life over that finite time period. While helping an orphan in India may have benefited Einstein, there were likely higher pay-offs closer to him, ways he could spend his time which would more likely benefit his short timespan for here. In this way, it is easier to understand the seeming insanity of Jesus':
“You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’ But I say to you, Love your enemies and pray for those who persecute you, so that you may be sons of your Father who is in heaven. For he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust. For if you love those who love you, what reward do you have? Do not even the tax collectors do the same? And if you greet only your brothers, what more are you doing than others? Do not even the Gentiles do the same?
If you are here on earth for but a short time before Death claims you, why spend too much effort enhancing the lives of those who won't likely enhance your own life? Now, we know that giving to others produces happiness, so this "enlightened self-interest" thing is more complicated than many indicate. However, every action has an opportunity cost; help too many orphans and you will spend miss opportunities to build relationships with other scientists and unlock the secrets of the universe. This may be more important than we think; see Richard Hamming's You and Your Research, in which he emphasizes the importance of dedicating one's time to science, in and out of the lab.
Anyhow, Einstein is describing the development of a story which arises from his will, that can fit into the constraints of a finite life on earth. This would be a god-like act if Einstein could have lived forever. As such, it sort of falls short; hence the common usage of the verb 'divine'.