You ought to renew the card and apply to naturalize to US citizenship - both of which can be done at the same time. Because the government is really in need of funds these days, they want to collect where they can, and motivated permanent residents seeking US citizenship are a source of revenue, even though there isn't an actual requirement to renew an expired lawful permanent resident ("green") card.
In fact, as part of naturalizing, one of the questions on the form is whether you want to change your name. You can actually have your first document with your new name your naturalization certificate if you chose to. Still, you should apply for a new green card because the one you've got is expired.
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Alice M. Yardum-Hunter, Attorney, Certified Specialist, Immigration & Nationality Law
State Bar of CA, Bd. of Legal Specialization