What does a female mosquito need from blood?
According to the Introduction to this paper:


  Attractiveness to biting insects is important in medical contexts, mostly in the dynamics of transmission of pathogens by mosquitoes that cause diseases such as dengue and malaria. Blood feeding is an essential part of the lifecycle for most mosquito species as it provides females with the proteins necessary for egg production.


I imagine that the blood protein is simply a rich source of amino acids. This is supported by the finding that an artificial meal consisting of bovine serum albumin can support egg production in Aedes albopictus.

Update following answer from Greg Thatcher

Clearly there is evidence that iron from a blood meal reaches ovaries and eggs, although the paper doesn't provide direct evidence that this iron is essential. However this made me think more deeply about the paper that I cite above about feeding bovine serum albumin as an blood substitute. Could this have also supplied a significant amount of iron? The artificial meal was BSA dissolved in a phosphate-buffered saline. The authors don't specifically address the problem of iron contamination. The main component of PBS is NaCl. If they used AR-grade NaCl then the iron content will have been very low (0.0002% in one example that I found online). However BSA itself can bind metal cations (see here for example) so the protein may have contributed some iron. The authors do quote the source of the BSA that they used but I haven't been able to access a data sheet that might shed light on the level of iron contamination.

Of course it is entirely possible that blood is an essential source of both protein and iron.