You can make it slightly more automated by using a Folder Action in Automator.
You should create a new folder just for screenshots as it might slow things down processing everything going into your Desktop folder.
Create a new Folder Action in Automator.
Add a Run AppleScript Action from the Utilities library.
Set the folder where you have screenshots saved.
Replace all of the text in the script with:
on run {input, parameters}
tell application "Finder"
activate
repeat with anItem in input
if name of (anItem) begins with "screenshot" then
set extension to "." & name extension of (anItem)
display dialog "Enter new name for " & name of (anItem) & ":" default answer "" buttons {"Cancel", "Rename"} default button "Rename" cancel button "Cancel" with title "Rename Screen Shot" giving up after 30
set newName to text returned of the result
if newName is not "" then
set name of anItem to newName & extension
end if
end if
end repeat
end tell
end run
This doesn't run until the thumbnail disappears from the corner of the screen as nothing has been written to the folder, yet. You can keep working as when it runs it will pop up the dialog asking for a name.
You can turn on and off Folder actions using Folder Action Setup (search for it with Spotlight). It should be set up already once you save the folder action.
I've never tested stacking up a bunch of screenshots, so you may have to be cognizant of the date-time of the file you are renaming if they somehow show up out of order.