As bummi already commented, having brown eyes can be expressed as braunäugig. You can then use this adjective as a noun:
Der Braunäugige lächelte.
It’s a bit unusual to define someone by their eye colour, though. Also watch out for blauäugig: While originally meaning having blue eyes in the same way as above, it can also signify naïve or gullible.
Sie sind sehr blauäugig an die Sache herangegangen.
The noun Blauäugigkeit actually exclusively denotes naïveté. (See Why is "blauäugig" used with a negative connotation? for the origin of this change of meaning.)
Generally speaking, -ig is a productive suffix that can be used to derive adjectives from nouns, and often meaning having … or exhibiting …. Examples include:
Maß – mäßig, Wasser – wäßrig, Sonne – sonnig, Suff – süffig, Gewicht – gewichtig.
As you can see, unstressed e is sometimes dropped from the noun, and the suffix may cause umlaut, though not always.
Extensions to the noun can in some cases be packed into the adjective, too, as in braunäugig (braune Augen) or außerhäusig (außer Haus).