Sometimes tears are not a sign of weakness. Even when they come easily and often...
Opposite Manly Tears are Tender Tears, where the character weeps more than most people because of the exquisite tenderness of their heart.
Tender Tears result from someone else's misfortune. If they feel any guilt over the situation — reasonably or unreasonably — Tears of Remorse will also occur.
Seldom a Single Tear, but it's possible.
A favorite trope of the Romantic movement, who regarded it as a mark of exquisite sensibility. Scorning them is a mark of a callous and hard-hearted soul; even those who do not cry themselves can appreciate the Tender Tears. Indeed, the tender character may explain the cause of the tears with such poignancy as to inspire Manly Tears in less tender characters.
Super-Trope of He Will Not Cry, so I Cry for Him and Tears of Joy.
Contrast Humanizing Tears, where crying makes a normally unsympathetic character more likable — Tender Tears is part of a consistently sympathetic character. This trope is used a lot on Prone to Tears and Nice Guy characters.