The well near which Hajar (Hagar) found water is Zamzam. This is one of the most well-known events in Islamic history and is directly connected to the rituals of Hajj and Umrah.
According to Islamic tradition, Prophet Ibrahim (AS) left his wife Hajar and their infant son Isma'il in the barren, uninhabited valley of Makkah by Allah's command. When their water supply ran out, the baby Isma'il began to cry from thirst. Hajar desperately ran between the two hills of Safa and Marwa seven times, searching for water or help. This act of running between the two hills is commemorated in the Hajj and Umrah ritual known as Sa'i.
After her seventh trip, the angel Jibril (AS) struck the ground near Isma'il (or the baby kicked the ground, in some narrations), and water miraculously gushed forth. This spring became known as the Well of Zamzam. Hajar contained the flow and the water has continued to flow ever since, making the location of Makkah habitable and eventually leading to the settlement of Arab tribes in the area.
The other options are names associated with water in Islamic tradition but are not the well of Hajar: Kawthar is a river in Paradise mentioned in Surah Al-Kawthar (108:1). Salsabil and Tasnim are both names of springs in Paradise, mentioned in Surah Al-Insan (76:18) and Surah Al-Mutaffifin (83:27) respectively.