I don't see the problem with what you're doing if you're replacing the money. It's a bit like knowing you're getting a £1K bonus in June, need to get a £1K deposit on a holiday now and borrowing the money.
As long as it goes back in June, you're using your bonus to pay for the holiday! If you kept the bonus that would be different.
And as your mum is all for you using the money she gave I don't see the problem.
The one thing you do need to be aware of is the temperature in Egypt. We went about 20 years ago and did Cairo, Luxor and the Valley of the Kings.
We had to be out of the hotel by 6am for the Valley of the Kings as the guide wanted us back mid morning because the heat is searing (it can go to 45 degrees sometimes) and it gets packed by about 10am. You will need to be prepared for going up and down stairs in that heat if you're going there.
Luxor is amazing but, again, you're outside in searing heat so be prepared. We did a light and sound evening there where you heard the stories of the Pharaohs and had a light display. Beautiful...and cool (thankfully) but the number of people! It was manic (think Wembley at FA cup final coming out).
Cairo was dusty and hot. Expect crowds at the pyramids and, again, go at a cooler time of day. Cairo temperatures can hit mid 30s quite easily (it's 31 degrees there at the moment apparently). So be prepared for walking in that sort of temperature.
And remember, loving pyramids at four years old is great. But your son will be in searing heat (it got too much for us sometimes). He may not be prepared for that - he'll be equating pyramids with the British Museum where we certainly don't see 45 degree temperatures and very early starts. It'll be exciting but certainly not easy for him. You need to think things through very carefully to keep him safe and happy.