Can I claim my cousins girl friend who I supported last year on my tax return?

She may fit the definition of a "qualifying relative" (even though she is not your relative), if (a) she is not anyone else's qualifying child - such as her own parents, (b) she lived in your house all year as a member of your household, (c) she has a gross income of less than $3,400, and (d) you provided more than half of her support during the year. Details are here:

Publication 501 (2007), Exemptions, Standard Deduction, and Filing Information