The Buddha something about what evidence you need to believe something. So how can one verify reincarnation?
In the Avatamsaka sutra, one of the powers that a Bodhisattva gains on the path is the ability to remember past lives.
While striving to attain Bodhi,
I will gain the knowledge of past lives in all destinies.
Which actually is a very Buddhist sort of proof, proof comes from personal experience. This is certainly not scientific proof, and for me isn't all that convincing.
This question here discusses what happens to Buddhism if we dispense with reincarnation: What if there is no rebirth?
I think the system holds up well without reincarnation. That said, the cosmology, with the six realms, King Yama & all make for a vivid way to discuss actions and consequences, more so that just talking in abstractions.
I know my answer might be perceived as silly but the argument I'm going to put forth really does convince me.
Martin Heidegger (one of the greatest philosophers of the last century, according to most) had a concept called Geworfenheit. This is translated as "throwness". It relates to the feeling we have that we feel "thrown" into this world.
If I have already experienced myself "thrown" into a world (this one) I think it's sound to believe that it might happen again!
If you want verification, then at least do yourself a favor and obtain the verification yourself, i.e. by doing insight-meditation (vipassana).
If you obtain verification by an external source, e.g. a scientist, another meditator, a Buddha, a teacher, a newspaper, the internet etc., then how can you know if its true? It could be false.
Go test it for yourself and obtain direct, experiential knowledge of reality.
If you do not know how to meditate or would like to learn it, Ven. Yuttadhammo has made a video series that might be beneficial for you.
Simply think that we haven't proved that.
Try to explain:
- Why there are animals if you see a life form in them why they can't be human.
- Why there are gods protecting the mankind, and how they are formed
- If we go to Heaven after we die, what happens next
- If every people is created equal why there is poor and rich
- For billions of years the world have existed when will the human kind end after all people go to heaven and hell. Is that the end?
- If math can explain the universe, the number line has also got 0 beside of negative and positive then what is that in real life. Is there anything.
- If you Don't do good and you won't do bad so what will be your destiny.
These are the usually explained stuff in Buddhism.
Nowadays net is full of evidences that claim reincarnation and children who insist they were someone in their previous life with the exact explanations of their personality in the previous life which sometimes are shocking. Because some of these children had many secrets in their previous life that is confirmed with those relatives and friends that know the secret. Just search YouTube for "Reincarnation".
"Mommy, Why am I Not a Boy This Time?" - a Story of Reincarnation https://www.youtube.com/watch?v=SQJXCBknbtQ
Reincarnation in America - Documentary https://www.youtube.com/watch?v=Z19SwRpJz0I