If you plan to use DC, you will need some sort of a "Switching Device" in order to have a functioning Transformer.
You could use a low voltage mechanical bell to accomplish this [a simple Electromechanical bell with an intregal switch, for use on AC or DC] and connect it in series with the Transformer's Primary winding.
Another option is to use a momentary contact switch, something like a "Code Tapper" used for Morse Code, so the Primary input can be as uniform as possible. This would take some practice!
These are a few of the simple methods you could use to create a varying input to the Transformer from a unidirectional and steady current source [DC].
Alternate methods using DC will involve Multivibrator circuitry to invert the DC into square wave AC.
If you are fluent with Electronics, a very simple setup can be made for around $10 - $20. You will need a few Bipolar Power Transistors, a few signal Transistors and Caps, Resistors, possibly a pot [to dial in the frequency] for the clock circuit.
Alternately, you can use an IC and a few external components for the clock circuit.
You will need a center tapped Transformer to be driven from the Inverter's circuitry.
Check the Inverter Schematics in the Tech Reference area [or are they in the Theory area??] for simple examples.
Your best bet would be to avoid all this baloney and power your Daughter's project from a 6 volt AC Transformer, plugged into a 120 VAC receptacle. Ground one line of the Secondary to stabilize the voltage to ground on the secondary circuit.
This could be used to power your custom wound Transformer, and the end result - to drive the Incandescent lamp.
Now to discuss the design criteria
If I was to cover the total amount of information required to design a Transformer, I would be typing for days and need to figure out just how to make up some extremely complex equations, using simple text only!!!
Even to cover basics would fill this thread bigtime!!!
I can give you something to work by, just for a very bare bones example.
A very simple and crude formula for turns per volt, using a somewhat high grade "E and I" silicon steel core:
For a Core Area of 0.5", the turns per volt (N/V)=
* 10 [at 60 Hz],
* 24 [at 25 Hz].
For a Core Area of 1.0", the N/V=
* 5 [at 60 Hz],
* 12 [at 25 Hz].
Conductor sizes will be according to the maximum load Volt-Amps [VA]. Figuring the Incandescent Lamp which will be driven from the Secondary may be something like 20 watts, figure at least 20 VA of Apparent Power flowing on both windings, then size the Conductors accordingly [I=E/P].
Wind your coils on a "Bobbin", then set them on the core. Weld the core sections together after all is setup OK. Lastly, pot the entire core/coil assembledge with Epoxy, to keep everything secure and solid.
Be sure to leave enough free conductor on the ends of each coil, so connections are not difficult. I suggest tinning the last 1/2" of each winding's lead conductor.
You may find on-line winding references, but I suggest checking your Public Library's Reference section for at least 2 different design manuals.
If you are good with Math [Algebra, differential and intregal Calculus], you will have no trouble with the formulas of design.
This project may be more than just a Father / Daughter Science project, it might be a nice bonding thing, and to encourage your Daughter with the Technical fields available!
Once you learn the indepth complexity of Transformers and similar Induction devices, you could very well become "addicted"
They are fascinating devices and once you create one with your bare hands that functions like it should, you will be hooked!
Even the ones that "Pop", loose smoke and cause "Mushroom Clouds" are not a total loss! Figuring out what went wrong is educational [and the explosions are impressive, too!]