I am probably going to forget a few but:
Not made up, legit spelling though less common is ok (Elinor vs. Eleanor). An actual name, not a noun.
Gender appropriate. I'm just not into gender neutral or the new fad of stealing boy names for girls.
Does have a nickname, bonus if there are several. Scarlett is nice but I'd hate for her to be Scar. Elizabeth is awesome because there are so many nicknames she can choose to go by. I cross off a name if a possible nickname will violate the gender appropriate thing. Love Charlotte, hate the nn Charlie.
I don't like it to be too popular or high on the SSA list. I also check Name Nerds' site to see the true popularity if there are spelling variations. But I make exceptions for classics like William and Elizabeth.
I usually compare 2011 popularity with that of 1880. I love 19th century and earlier literature, am a historian and so I love old names. I want it to have history.
I want it to make sense phonetically in English when the child learns to spell it. I like Gaelic names on someone else's kid but it's just not a good fit for me.
Meaning is really big for me. I like it to have significance to us as well. I might not discount Cecilia because it means blind, but I would discount a name that means demon or thief or something.
Has decent flow, but if there's greater significance then that outweighs the flow.
DH is really into honoring family. I like that for his side, but there's no one on my side I want to honor. So this is the least of my rules.
ETA: I did forget one! I would prefer no one in my circle of family/friends to have the name or something very close to it, unless I'm picking a popular name. I know this is insane and futile because circles evolve and there's no guarantee to keep it that way. But I was traumatized as a kid from this. I won't think twice about knowing another Elizabeth but I will think twice about knowing another Eloise.