I would "use" your ethnicity on your applications if it has any relevance to making you who you are now. Have you developed any connections to the Hispanic/Latino community? Have you done any public service work? Have you mentored kids who didn't come from a wealthy white background?
Based on what you've said, I think it'd be somewhat disingenuous of you to now use that background solely for gain. However, given the low numbers of Hispanic/Latino law students, using it might get you some mileage.
If you're interested in reclaiming this part of your heritage, I'd suggest you get involved with a related group while you're in school.
To me, the idea of diversity is getting people who aren't all from the same background, to bring in diverse ideas, and to train people who'll eventually work in a variety of places and positions. I had a friend who was first-generation Mexican-American, came from a poor background, had primarily Spanish-speaking parents, and, during his second year, convinced himself he wasn't qualified to be a law student and dropped out. That's a loss to the legal community. I'd rather have a guy like that, who definitely brings a whole different outlook to the situation, in a class than one more white kid whose upbringing mirrors half the class. (And, having been in classes with the guy...I certainly think he was qualified to be where he was.)
Anyway, sorry, but that's my little diversity soapbox for today.