Even after reading your post, I initially read m-i-n-j. Without the provided info, it could just be read as m-i-j. Separating the 'm' from the 'n' will most certainly help this concept if you intend to continue with it.
Do be aware that in the UK, reading as minj could give the design certain distracting connotations. The word 'minge' over here is a slang word for (how do I put this tastefully?) a ladies special place.

That aside, I'm definitely more drawn to the first one. The curves and circular tittle make it much more approachable.
As you're simulating a real brief, it might be best to delve a little deeper into what Ming is all about. What style of artwork does he/she produce? Is it a graphic style or more of a fine art style? The more you know about the client, the more you can suit the design to them.

I was worried it could be construed as "minge" when I started. Do you have any suggestions for how to make it look more like the tail of a g than a j? Here is a sample of the tail changed to that of a g.

In my mind, I think that sharper curve of the tail looks odd when compared to the shoulder of the m. What do you think?

I don't read it as Ming, but it could certainly be used as a logo that relates visually to her name but doesn't literally spell it out.

Regarding the dot over the i, you might try a square one with radius corners.

That's what I was going for, James. Not necessarily a literal m-i-n-g, but a logo that would be indicative of the name.

I like your suggestion about the rounded square for the tittle. I'll have to sketch up a version like that.

Do you have a preference as to the tail? I think that is what is giving me the most trouble right now.

the g tail is better - but i see this logo as a closed fist coming my way... minge karate dojo anybody?

it isn't feminine at all - but am i only assuming ming is a girl? - think i have been distracted already...

Side note: I cannot get the image of a minge karate dojo out of my head, haha.

Try creating an ear such as one would usually find on a binocular g.

I just dont like the design concept. It's abstract, and it's not. You know what I mean?

I would try to make it even more abstract.

Tim is right on. Either it has to be correctly legible (and there's no
way this reads anything close to "ming") or it has to be a mystery.

hhp

totally abstract... Plessey electronics logo...

http://a0.twimg.com/profile_images/617981534/plessey_logo.jpg

Hmmm, I wouldn't say Plessey's logo is totally abstract. To me, it is quite clearly a typographical device that spells "Plessey" using the visual style of an oscilloscope trace, even adhering to the fact that an oscilloscope trace cannot go 'backwards' (as it is a timebase).

What about adding another hump on the M and close the g shape slightly so it's not as open, but still keep it disconnected.