Excel history goes back to 1982 with the software Multiplan, which used R1C1 style references. There was a conscious decision when Microsoft Excel was created to be compatible with Lotus 1-2-3, the "killer app" for PCs in the 1980s. Lotus had chosen A1 style notation, and used the dollar signs to distinguish between absolute and relative references. Lotus had copied Visicalc, an even earlier spreadsheet program.
For Excel to be successful, that is, to be profitable, Microsoft had to attract Lotus 1-2-3 users. So A1 cell notation, with the dollar sign as a reference modifier, was used because Lotus had used it.
The actual choice of the dollar sign glyph? The original spreadsheet programmers needed a singe glyph (symbol) that is easy to type, found on the (U.S.) keyboard, and the symbol has to be one of the 128 ASCII characters, the only computer character set available in the seventies and eighties. Other computer languages had already made engineers and scientists familiar with the dollar sign as a modifier for variables and references in programs. The spreadsheet designers simply imitated a notation style with which they were familiar.
From a coding perspective, R1C1 references are much easier to calculate "in your head." Who can remember that the cell 1,017 columns to the right of A1 is $AMC$1? Or is it $AMB$1 or $AMD$1?