Whats the name?

[shanreagh]

Looks like Tabil, Sabil or Jabil to me.
There are Tabils in Warwickshire

[sparrett]

Possibly Dalziel.

I think I read somewhere that the z is silent?

Possibly written as it was heard and the z omitted?

There seem to be a couple of families of that name in the area.

The woman looks at least into her 30's to me.

Sue

[Whipby]

Trying to think outside the box - could it be Dobie or Davie?

[maddys52]

TALMIE / TALLIE ?

I don't think it's a "b" in the middle, but I could well be wrong!

[Lisa in California]

Could the first letter be a T, like others have mentioned?  If you try to recreate the letter, starting with a downward stroke, to me, it flows naturally and appears to be a fancy T.  Perhaps it has that weird little “wing” at the end to balance out the letter?

Added: the writer added little flourishes to all of the capital letters.

[Lisa in California]

I thought if I could find W. J. Wright’s writing if we were lucky it might include enough letters that we could figure out the lady’s name.

Unfortunately (providing that I found the correct man in 1911), the writing on the census form doesn’t appear to be a match to the writing on the photograph.  Providing his details, in case anyone wants to view the image:
William James Wright, Photographer.  Born c1864, Isle of Wight.
1911 North Croyden, Surrey census

I also checked a couple of family trees (Anc*try), hoping for more samples of his writing, but his signature on his childrens’ marriage certificates appears to match the 1911 census.  There were no images in those family tree “galleries” (I was hoping to see a couple of his photographs with the same style of writing as the writing found in this post). 

So much for my thinking outside the box. 

[Girl Guide]

I'm inclined to go for Tabil.  Attached is the name in a different shade.

[arthurk]

Could it be spelled phonetically, and thus a slight misspelling?

The 1901 census has an Emmeline De Bil, age 17, in Temperley Road, Balham (RG13/475 fo83 p39) - that's not a million miles from Upper Norwood.

The De Bil family consists of parents and two daughters, all of whom are French, and the father is a language teacher. It made me wonder - might the photographer have visited schools and set up a temporary studio to capture all the pupils?

If I'm right and this is Emmeline she'd be about 15-16, which I think may be possible.

[peakoverload]

Thank you all for your suggestions and research, it is very much appreciated. There's certainly some leads to follow up on. I imagine that the name was written on the card by one of W. J. Wrights assistants and not a family member (a family member is less likely to use Miss), but what I'm not sure of is whether this photo was owned by the sitter or whether it was kept by W.J. Wright as a reference for it further prints were required. If it was owned by the sitter than I would think it's more likely that the name would have been corrected if misspelt. Definitely further research is needed as I'm very intrigued to find out more about this person.