Senator Griego represented District 39, a large district that includes parts of six counties. Each of those county commissions has the right to recommend a replacement to Governor Martinez. Because Senator Griego was a Democrat, Governor Martinez would presumably like to quickly appoint a Republican to fill the seat.
Earlier this week, a petition was filed with the New Mexico Supreme Court asking for a writ of mandamus to compel several county commissions to hold emergency meetings to nominate replacements. (Hat tip to Joe Monahan for the link to the petition).