The translation of Paseo de los Tristes is "walkway of the sad people". This is because there is a cemetery just above the Alhambra and there is a pathway from Paseo de los Tristes called Cuesta de los Chinos (hill of the little stones) which goes up to the Alhambra. The funeral processions used to follow this route up to the cemetery. Apparently the square got its name during a yellow fever epidemic at the start of the nineteenth century.
Nowadays the people who sit at the open air restaurants in Paseo de los Tristes are far from sad. There is a choice of several outdoor restaurants. The square is directly below the Alhambra and has wonderful views. It is a very pleasant stroll to go along the river darro from Plaza nueva and end up at Paseo de los Tristes.
The route up to the Alhambra via Cuesta de los Chinos is a good shortcut from the Albaicin up to the entrance to the Alhambra. It is a pedestrian only path and is fairly steep, it could be the most picturesque way to go up to the Alhambra. To go up there cross over the bridge at the end of the paseo and it is straight ahead. By the way if you go left rather than straight on it will lead you on a short walk to the Fuente de Avellano. (the spring of the hazelnut tree)
There is a page with infor about the bars and restaurante at Paseo de los Tristes here