express condolence
唁 centers on expressing condolence, specifically the act of verbally offering sympathy or mourning words to someone who has suffered a loss.
唁 combines 口 (mouth) with 言 (words), suggesting the act of speaking words of condolence. The exact historical development is uncertain, but the component choice clearly relates to verbal expression of sympathy.
A mouth 口 speaking words 言 of comfort: when someone is grieving, you use your mouth to offer words of condolence.
For ゲン, imagine a gentle person named Gen offering condolence: Gen -> ゲン, and his kind words come from the mouth 口 and words 言 of 唁.