As in the case of  many Welsh surnames this was originally a forename. In Old Welsh it was spelled "Grippiud". According to Morgan and Morgan, this would have become Griffudd and then Gruffudd on the way to its modern Welsh form of Gruffydd. 

Medieval scribes of English or French origin spelled the name "Griffith" and this form ultimately gained acceptance. The form Griffin was also commonly used, especially in Latin documents where it was better suited to take Latin case endings. (A recent report from America of "Griffing" may well indicate early emigration from Wales.)

When the name became used as a surname a final "s" was often added, giving us the version most commonly found today. 

Many variants have been noted. A glance at the 2000 telephone directory for Newport and South East Wales reveals the following:
Griffiths - around 700 entries
Griffin - around 120
Griffith - 25
Griffths - 4
Griffis - 2
     

Older documents provide Gryffyth, Gryffyd, Gruffuth (13th century) and versions such as Griffyt, Graffit, Griffit, Griffeth, Greeffethes, Greffet and even Gripthis! The authorities think that names such as Gittoes and Gittins derive from the pet forms of Gruffydd, namely Guto and Gutyn.  

Welsh surnames