I gave this a try with my comment:
ብርሃን bǝrhan is from በርሀ bärhä "to be bright". This is the source of በራ bärra too. ብሩህ bǝruhǝ is probably related, since it has b and r at the start. برق is either the verb baraqa "to produce lightning" or "to shine, glitter, sparkle, flash", or the noun barq "lightning", and comes from Proto-Semitic *bariḳ-. This doesn't seem to be related to the Amharic words, based on the relevant Wiktionary entry. That's all I got unfortunately.
Then I asked K.W. in a comment on Quora, and he replied:
The Wiktionary page برهان - Wiktionary seems to help out with this a lot :).
From the root ب ر ه ن (b-r-h-n), interrelated or derived from the roots ب ر ه (b-r-h) meaning "to make white", "to clear up", "to restore to a state of youth or innocence" and ب ر ء (b-r-ʾ) meaning "to be free from wrong", "to be absolved from wrongdoing", "to be without blame", "to be innocent".
From Ge'ez ብርሃን (bərhan, “light, brightness”), from በርሀ (bärhä, “to be bright, to be clear”).
From Old South Arabian 𐩨𐩧𐩠𐩬 (“testimony”).
Also, for the link below, it says the following about “Ethiopic” (another name for Ge’ez). Nevertheless, the source below is still useful: Semitic Words in Egyptian Texts of the New Kingdom and Third Intermediate Period
Also, for one small section in the book below, it speaks to some kind of link between the Arabic بر (or برهان in the link/screenshot below) and ብርሃን. Unfortunately, the page before it wasn’t included in the preview, which would have been more clarifying. However, the screenshot was enough for me to recognize a kind of link in some way. Recall again the reference to Ge’ez as “Ethiopic": Ethiopia and the Bible
I hope this all helped you out.
𐩨𐩧𐩠𐩬 transliterates to brhn.
I wonder if there is a root b-r from which all these words derive… or if b-r-' or b-r-q could have somehow dropped their third element to form a word ber…
Bright and brilliant probably have nothing to do with the above words, since bright is from PIE *bhereg- and brilliant comes, with some steps I skip, from Italian brillare, from Greek βήρυλλος, from Prakrit विरलायते viralāyate, from Sanskrit वैडूर्य vaiḍūrya, a Dravidian loanword (wait, Dravidian veḷur was borrowed into Persian and Arabic? As what word?), and I'd say it's far more likely for the above Afro-Asiatic words to be related to ber than for the English words to be.