No special word, but "yield your seat" or "concede your seat" would be slightly more formal alternatives to "give (up) your seat", or with a slight change in tone, "offer (someone) your seat"
The seats designated as such are called "priority seats"
The tradition of who gets to sit in a priority seat varies from culture to culture. In some cultures it is mostly for the elderly, in others it is for children, disabled people, pregnant women, or people with young children. In some cultures it is just "those in need".