tears, sympathy
涕 centers on tears as a physical expression of emotion, and by extension the sympathy or shared feeling that tears can evoke.
涕 combines 氵 (water) with 弟, which likely contributes the sound. The water radical suggests tears as a liquid expression of emotion.
Water 氵 beside 弟: imagine a younger brother 弟 shedding tears 氵 of sympathy for someone else.
For テイ, picture a tearful person handing you a tissue with the word 'TAY' written on it: TAY -> テイ, and the tissue catches the tears.