You can use this event code for the particular worksheet you want to have this functionality (it assumes the "specific cell" that will be monitored is cell A1, change as needed)...
Private Sub Worksheet_Change(ByVal Target As Range)
On Error Resume Next
If Target.Address(0, 0) = "A1" Then Target.Parent.Name = Range("A1").Value
If Err.Number Then MsgBox "Your entry is not valid for a sheet name!"
On Error GoTo 0
HOW TO INSTALL Event Code
If you are new to event code procedures, they are easy to install. To install it, right-click the name tab at the bottom of the worksheet that is to have the functionality to be provided by the event code and select "View Code" from the popup menu that appears. This will open up the code window for that worksheet. Copy/Paste the event code into that code window. That's it... the code will now operate automatically when its particular event procedure is raised by an action you take on the worksheet itself. Note... if you are using XL2007 or above, make sure you save your file as an "Excel Macro-Enabled Workbook (*.xlsm) and answer the "do you want to enable macros" question as "yes" or "OK" (depending on the button label for your version of Excel) the next time you open your workbook.