Blepharitis: This is an inflammation of the eyelids that can cause swelling, redness, and drainage. It's a common condition and can be caused by a bacterial or demodex mite infestation, making it a likely diagnosis for eyelid swelling and drainage.

Conjunctivitis (Pink Eye): This is an infection or inflammation of the outer membrane of the eyeball and the inner eyelid. It can cause redness, itching, and discharge, which might be perceived as drainage from the eyelid.
Hordeolum (Stye): A localized infection or inflammation of the eyelid margin involving hair follicles of the eyelashes or meibomian glands. It presents as a painful, swollen nodule on the eyelid, which could be accompanied by slight drainage.
Chalazion: A small bump on the eyelid due to a blocked oil gland. While it's usually painless, if it becomes infected, it could cause swelling and slight drainage.

Orbital Cellulitis: An infection of the soft tissues surrounding the eye, which can cause swelling, redness, and pain. It's a serious condition that requires prompt treatment to prevent complications like vision loss or spread of infection.
Preseptal Cellulitis: Similar to orbital cellulitis but less severe, involving the eyelid and skin anterior to the orbital septum. It can still lead to serious complications if not properly managed.

Dacryocystitis: An infection of the lacrimal sac, usually caused by a blockage of the nasolacrimal duct. It can cause swelling, redness, and discharge in the area of the eyelid near the nose.
Angioedema: A condition characterized by the rapid swelling of the dermis, subcutaneous tissue, mucosa, and submucosal tissues. It can cause eyelid swelling and, if involving the mucosa, might result in slight drainage, though this is less common.
Eyelid Malignancies: Though rare, certain types of skin cancers (like basal cell carcinoma or squamous cell carcinoma) can present with swelling and possibly drainage if they become ulcerated or infected.