There are several lookout towers in Budapest where you can take a good panoramic photo, and I think one of the best gifts for your friends is a beautiful shot shared with them of the places you have been to.
To start with the city center, you should definitely get to the top of the Basilica on the Pest side, either being sporty and climbing the 364 stairs, or taking the elevator (see the link for more details on the opening hours). The Pest side is totally flat, so this could be the topmost part of the eastern part of Budapest.
As for the Buda side: the Fishermen’s Bastion is a typical choice since you can get a clear view of the River Danube with the Szechenyi Chain Bridge, the Parliament, the Gresham Palace, etc. Another frequented spot on the Buda side is around the Statue of Liberty on top of the Gellert Hill. I think the top of the Tunnel (right overlooking the Chain Bridge) is also a good spot.
Outside the inner city
Outside the city center you can take panoramic photos on Normafa, for instance, or take the Children’s Railway and get off at Janos
hegy, at the Elisabeth Lookout Tower (in Hungarian Erzsébet kilátó). Although it’s a beautiful memorial, admission is free and kids also love it very much.
Another option on the route of the railroad is to go to the Szechenyi Memorial Lookout Tower, which is slightly lower than the Elisabeth. The ride on the Children’s Railway itself also has a panoramic part.
Check out the lilac camera icons on the Budapest Tourist map to find your panoramic photo places easily: