It's not entirely clear if the pattern @[name](id info)
is fixed, but if so, the following should work:
Dim input = "blah blah blah blah @[Dave Battersby](Person:292) blah blibbidy blah"
Dim output = Regex.Replace(input, "@\[(.*?)\]\(.*?\)", "$1")
This searches for the full pattern, capturing the name part, and replaces the matched text with just the name.