Its worth mentioning at this point that the Snap-Circuits system the O.P. is using supports the
PICAXE-08M2 MCU, which can be programmed graphically in a way that's more accessible to younger children.
See
https://www.elenco.com/wp-content/uploads/2011/08/SCXP-50.pdf for the manual of the now discontinued SCXP-50 kit that shows how to use a PICAXE-08M2 with Snap-Circuits. To get the essential parts you need to use a PICAXE-08M2, see the Snap-circuits spare parts page:
http://cs-sales.net/sncirepa.html (Edit: stupid site doesn't allow deep links). Look for "PICAXE Micro IC in Socket # 6SC U21" or if you already have a suitable PICAXE-08M2 "8-pin IC Socket ONLY with Micro Marking # 6SC U21S"
Its worth noting that a PICAXE chip needs a couple of resistors on its serial input pin (see Nuts & Volts article link below) to prevent noise interrupting the program. For it to work properly the #6SCU21S PICAXE socket snap must include them internally. If you wanted to use a generic "Eight-Pin IC Socket # 6SC ?U8" snap, you'd need to make a little carrier board with a socket for the actual PICAXE chip, those resistors, and an 8 pin header to fit the DIP socket on the 6SC?U8.
The official PICAXE system download cable is terminated with a 3.5mm stereo jack plug, and the Snap-Circuits SCXP-50 download cable with snaps on the end ("Cable USB to Snaps # TL SCXP" on the spares page) is rather expensive at $40 so you'll probably need a programming jig or adapter.
Probably the easiest/cheapest option (with a competent electronics tech in the house) would be to DIY the programming cable - see
http://www.nutsvolts.com/magazine/article/july2014_Tylerand buy orange, yellow and black snap to snap jumper wires (search spares page for "jumper wire") to hack up and solder to the DIY programming cable board. However, one could just get the official PICAXE cable, and the Snap-Circuits "Audio Jack on Snaps, Horizontal # 6SC JA" + those three jumpers and use the 6SCJA as an adapter.
Once you've jumped through all those hoops, you can download for free all you need to program PICAXE MCUs in BASIC or graphically using flowcharts from the official PICAXE website:
http://www.picaxe.com/Software. There's also support for using it with MIT Scratch.
N.B. To avoid frustration and resulting lack of interest you need to have 'all your ducks in a row' before putting this in front of a child - All hardware assembled and tested, all software installed and a basic 'blinky LED' hello world program running on the PICAXE-08M2, and enough familiarity with the PICAXE toolchain that you can demonstrate 'drag & drop' flowchart programming to the child without fumbling it.
Edit: I found the Snap-Circuits PICAXE USB to snaps cable on the spare part page, + some problems with the links to parts, so a lot of changes above.