Electronic – Vacuum tube LED flasher

If you don't know anything about the transformer then just give it a try and see what you get - a blocking oscillator can be made with just about any transformer, so the chances are pretty good it will work first time. There are a couple of variations of the "Joule Thief" (blocking oscillator) circuit, here is the most basic:

The transformer does not have to be the hand wound toroid shown, and the transistor can be any general purpose NPN bipolar. Note the windings must be connected in opposite polarity (dot convention) You may have to vary the resistor a bit to achieve desired operation (frequency should increase with resistance)
As it happens, I wrote a blog entry a while back on the blocking oscillator circuit used in a buzzer, which was itself part of a cheap "mole repeller". It discusses the operation in a little more detail and shows an alternative topology, so may be of some use to you (you want the bit under the heading "what frequency does the buzzer run at?")

EDIT - to clarify about the connections and some caution:

Since you have a centre tapped secondary, you should be able to use this assuming it's a standard centre tap - the centre tap would the topmost connection to V+ in the circuit above, then one lead to the resistor, and the other to the transistor collector.

In all cases, make sure you have a load (e.g. your LED) connected before you apply power - otherwise you may get pretty high voltages developing at the transistor side, which will damage it. Also, as Acme Fixer points out wisely, if it's a mains transformer you will get high voltage developing at the primary winding, so make sure these are insulated well. The currents will be low from a much stepped up and half dead battery, but could still give a nasty zap so treat with caution.

As Dave said, you don't have to actually run the tube in a real circuit to get the visual effect. All you have to do is power up the heaters.

Your particular tube actually is two tubes in one package and has two heaters. Each heater is 6.3 V. You can put 12.6 V accross pins 4 and 5, or 6.3 V accross both pins 9-4 and 9-5. In the latter case, you connect one side of the 6.3 V supply to pin 9, and both pins 4 and 5 to the other side.

Don't worry about the exact voltage that much. These things were intended to be run directly from a 6.3 V output winding of a power transformer. It will be OK with variations of this 6.3 V due to input line voltage variations. Get a "6.3 V" transformer intended to run from whatever power you have in your location and the tube will be fine.

You do have to make sure the transformer is rated for the total current of all the heaters together. For example, if you are powering 5 heaters that all take 6.3 V at 300 mA, then the tranformer output must be rated for at least 1.5 A.