The two double quotes are a way to indicate that the double-quote is actually part of the string, not a delimiter.
You set your pattern accordingly.
Text = "<span title =""Target text to be extracted"" custom_id-""5"">SomeOtherText"
Pattern = "title\s?\=\s?""(?<title>.+?)"""
Title = System.Text.RegularExpressions.Regex.Match(Text, Pattern).Groups("title").ToString