Azure blob storage - auto generate unique blob name

You can use a Guid for that:

string blobName = Guid.NewGuid().ToString();

In addition to what Sandrino said (using GUID which have very low probability of being duplicated) you can consider some third-party libraries which generate conflict-free identifiers example: Flake ID Generator

EDIT

Herve has pointed out very valid Azure Blob feature which should be considered with any blob names, namely, Azure Storage load balancing and blobs partitioning.

Azure keeps all blobs in partition servers. Which partition server should be used to store particular blob is decided on the blob container and the blob file name. Unfortunately I was not able to find and documentation describing algorithm used for blobs partitioning.

More on Azure Blob architecture can be found on Windows Azure Storage Architecture Overview article.

There is nothing that generates a unique name "on insert"; you need to come up with the name ahead of time.

When choosing the name of your blob, be careful when using any algorithm that generates a sequential number of some kind (either at the beginning or the end of the name of a blob). Azure Storage relies of the name for load balancing; using sequential values can create contention in accessing/writing to Azure Blobs because it can prevent Azure from properly load-balancing its storage. You get 60MB/Sec on each node (i.e. server). So to ensure proper load-balancing and to leverage 60MB/Sec on multiple storage nodes you need to use random names for your blobs. I typically use Guids to avoid this problem, just as Sandrino is recommending.