“Ihai” (位牌, いはい) is a wooden plate with a posthumous Buddhist name on it to honor the deceased. It originates in the convention of Confucianism where the information of the dead person includes the rank and name.
This is called a “Kamidana” (神棚, かみだな). It is an altar to enshrine the god of shintoism, and is kept in different places like your home or in an office. When you put it in your home, you normally pray for your family’s safety and health.
Normally, Daruma’s eyes are blank. Right before an exam, an important match, or a politician running for a race will pray for success and paint the left eye. Then once this wish has come true, they will paint the right eye.