You have to standardize on some common granularity for strikes/expiration dates so that each symbol has sufficient volume so that the market would not get too sparse. The theory does not require it, but when you are building an exchange, I suppose that'd be of a practical concern to you. You want interchangeability in derivative contracts and don't want each individual contract to be a snowflake.

There are "mini" options that control 10 shares, for example, but they usually have higher bid-ask spread reflecting the higher cost of pricing accurately/transaction fee/lower liquidity.

100 shares is considered a "lot" and is a standard size. True, an investor can own fewer than 100 shares, but every so often a company will make an "odd lot offer" forcing investors to either sell their shares or buy enough to hold at least 100.

Another issue is that out of the money contracts for a single share would frequently trade for pennies, which would greatly increase transaction costs and make the market less efficient. So you would have to standardize on some amount that is large enough to avoid such inefficiency, and 100 is a convenient number for mental arithmetic and reduces confusion. In theory you could have standardized on a larger number like 1000, which is equally convenient, but that would have priced out retail investors who may not own 1000 shares of a company whose share price is in the double or triple digits (which is common) and reduced the liquidity of the options market (which is already not very liquid).

Different derivatives have different lot sizes iirc. Options contracts for listed equities have a lot size of 100. Once options markets became more standardized in 1973 with CBOE introducing call options, the default lot size became 100.

From /u/ins2be on reddit:

"That's what the OCC sets it at in their bylaws.

Unit of Trading

(5) The term “unit of trading” in respect of any series of options or futures means the number of units of the underlying interest which have been designated by the Corporation as the minimum number to be the subject of a single option contract or single future in such series. In the absence of any such designation for a series of options or futures in which the underlying security is a common stock the unit of trading shall be 100 shares."