Why do you say "to place a call"?

My first guess is "place a call" originated from when the operator would physically have to plug the wires on a switchboard to make a call.

From HowStuffWorks:

Think back to a time when a human being worked in a town's central office. The phone company would build the central office in the middle of town, and then run a pair of copper wires to every home (see How Telephones Work for details on the wiring). The operator -- let's call her Mabel -- would work in the central office. Mabel would sit in front of a switchboard, and on this switchboard would be a collection of sockets -- one socket for each of the phones in town. When you wanted to place a call:

• You would pick up your phone.
• A light above your phone's socket would turn on.
• Mabel would plug a jack into your socket and ask you who you would like to talk to.
• Mabel would then plug her jack into the receiving party's socket, send a ring signal down the line, and talk to the person who answered.
• Mabel would then plug in a wire between your jack and the receiving party's jack to connect the two of you together.
• When she saw the lights go out above your jacks, Mabel would remove the wire connecting the two sockets.

Electrical review, Volume 45 from 1904 talks of placing a call in front of an operator. This corresponds with the switchboard lights turning on in front of Mabel:

Wires should be led from this trunk plug to a desk located outside of the operating room, and having facilities for testing the called line after the plug is inserted, and listening upon it without closing it, so as to place a call before the operator if it is not in use.

So when the caller picks up the phone, they are placing something in front of an operator (some indicating lights) to indicate you wish to make a call. The operator then places some wires in plugs to initiate the call. This gives us at least two senses of things being placed.

I'm not sure, but I wonder if it's a holdover from days when callers needed operator assistance to place a call.

Because callers needed help calling a place, the language place a call would naturally evolve. If I picked up the phone, and my friendly neighborhood operator was on the other end, "I'd like to you to help me place a call" seems like a perfectly natural thing to say, rather than "I"d like you to help me call a place." (That's just a theory, but the blue line on the graph would seem to support it, as it coincides with the spread of the telephone into rural areas.)

As to why make a call is so much more widely used in the Ngram, not all references to make a call revolve around a telephone. A referee can make a call on the field. Because make a call is an idiom for make a decision, a board of directors can make a call about a new stock option.