Where the association comes from
Monday's Sanskrit and Hindi name, Somvar, comes from "Soma" — a name for the moon, which Shiva wears as an ornament in his matted hair. That lunar connection is the traditional root of pairing this particular day of the week with Shiva worship, alongside his association with coolness, stillness and the ascetic, moon-like temperament in contrast to the sun's intensity.
How the day is typically observed
Many devotees keep a Monday fast — either a full Somvar Vrat (avoiding grains, eating once a day) or the more specific Som Pradosh observance when Monday coincides with the Pradosh period (roughly the hour and a half around sunset on the 13th day of the lunar fortnight, itself already associated with Shiva). It's also common simply to visit a Shiva temple, offer Bel leaves and water to a Shiva lingam, and chant Om Namah Shivaya through the day, rather than keeping a formal fast.
Sixteen Mondays (Solah Somvar)
A specific vow, Solah Somvar Vrat, involves observing sixteen consecutive Mondays with fasting and Shiva worship, traditionally undertaken for a specific wish — commonly related to marriage, family wellbeing, or resolving a significant obstacle — concluding with an udyapan (closing ritual) on the seventeenth Monday.
Why a weekly rhythm helps
Beyond the specific mythology, dedicating one fixed day of the week to a particular deity is a practical devotional tool: it gives an otherwise open-ended practice a rhythm, similar to how Tuesdays and Saturdays are associated with Hanuman, or Fridays with the goddess. You don't need a special occasion to deepen your practice — the calendar already offers one, every week.