Why did the Mathematica Language choose term rewriting instead of the Lambda Calculus as its basis?

The short answer is that the Mathematica Language did not choose anything. It was Stephen Wolfram back in 1979 when he started working on his own system after he reached the boundaries of Maxima which was his program of choice then.

There is a very interesting talk about this which is called How Mathematica, Wolfram|Alpha & the Wolfram Language Came to Be. In this talk, he described some of the reasons why he designed it the way he did. You might want to watch it from minute 24 when he talks about Algy - the algebraic manipulator which later became SMP and finally Mathematica.

Here is the probably most related part, freely transcribed by myself:

I knew most of the general-purpose Algol-like languages and as well as languages like Lisp and APL and so on at the time, but somehow they didn't seem to capture sort of the things that I wanted my system to do. So I guess what I did was what I learned to do in physics which was I tried to sort of drill down to find kind of the atoms; the primitives of what was going on in all these computations that I wanted to do.

I knew a certain amount about mathematical logic and the history of attempts to formulate things using logic and so on, even if my mother's textbook about philosophical logic didn't exist yet, but the history of all the effort of formalization I was quite aware of through Leibnitz, Hilbert, [...]

Back in 1979, I was sort of thinking about this kind of thing and that led me to design the design that I came up with that was based on the idea of symbolic expressions and doing transformations on symbolic expressions.

This all does not sound to me as it was an active decision to create a term rewriting system but rather, Wolfram wrote down the specifications of how he thought an expression manipulator should be designed. When we look at it now, it seems clear that it is of course a term-rewriting system, but maybe it wasn't so clear back then.

A somewhat ahistorical or anachronisic take on "why" is given in Bruno Buchberge's article Mathematica as a Rewrite Language, which shows what can be done with term rewriting as developed within Mathematica. In particular,

... this gives a versatile and practically attractive potential for providing all the quantifiers one would want to see in a truly mathematical language.