We are in the process of nutting out the design guidelines we would like to use in our development team and got into a discussion today around how ASP.NET controls should be
I'm not sure of the guidelines regarding ASP.NET, but in the book Framework Design Guidelines from Microsoft, there are several best-practice guidelines about naming of class members. Since ASP.NET Controls in most cases result in a protected field of the appropriate type, I consider these naming guidelines to apply for ASP.NET controls as well. In fact Code Analysis does not differentiate on Control reference fields and other fields.
These guidelines recommend using a naming scheme that implies the logical use rather than a type-descriptive variant. There are several reasons for this. The prefix is implies a type to the developer that might not be correct due to later changes. It adds an extra step in code maintainence. If you change your Button control into a LinkButton control the name also needs to be changed to correct the prefix.
For that reason I would call the control FirstNameEdit etc...