Cause of Appearance of Blood Vessel in the Eyelid?

There are many reasons that were listed below as well as genetics, thin skin and sun exposure. Treatment is a high volume center with experience is low risk and use of a eye protective shield is key when using lasers to obliterate these. 

but your own body genetics play the largest role here.

it is hard to diagnose your condition without some photos but there are treatments for veins around the eyes as well as those on the nose.  increased pressure in the veins can cause this and  you will need a full medical assessment to diagnose your condition.

Cheers

Yes,all the above. Extra flow to the blood vessels can increase the size and visability of blood vessels.Cool compression, mental relaxation and a trusted friend to discuss your sadness/ prevention of crying episodes will all help.Best. Dr Brecht Seattle, Wa 

Any ot the above things you mention could lead to increased vasculature or blood vessels around the eyelids. Anything that increases pressure in the blood vessels can cause exisitng veins to enlarge or cause new veins to form. This can even include straining from weight lifting or just coughing.
These veins may decrease in size in response to less pressure, strain, or trauma but they will not go away on their own.
Large veins can be treated around the eyes using longer wavelength lasers such as the 755 nm alexandrite or the 1064 nm neodynium:YAG lasers with long pulse durations. When treating veins on the upper eyelid or near the eye, a metal intraocular eye shield that is inserted under the lid must be used to protect the eye from any laser injury.