Define function that behaves almost identically to Mathematica function

If you want to constrain it to only options from ListPlot, you could use OptionsPattern in combination with FilterRulesand Options.

myListPlot[data_, opts : OptionsPattern[]] := 
 ListPlot[data, GridLines -> {None, {data[[1]]}}, 
  FilterRules[{opts}, Options[ListPlot]]]

which results in:

myListPlot[data, PlotStyle -> Red, Joined -> True]

The usual way to define a Wolfram Language function that takes n arguments and an arbitrary number of options is like this:

f[arg1_, ..., argn_, opts___] := ...

A little bit of pattern matching background (taken from the WL reference):

• _ any single expression
• x_ any single expression, to be named x
• __ any sequence of one or more expressions
• x__ sequence named x
• x__h sequence of expressions, all of whose heads are h
• ___ any sequence of zero or more expressions
• x___ sequence of zero or more expressions named x
• x___h sequence of zero or more expressions, all of whose heads are h