How to make this ping test with timer more efficient?
Using Application.OnTime is a very neat way of implementing your timer, but I have a hard time figuring out how "pings servers every 15 mins or so" translates to Now + TimeValue("00:00:01"). Note that since VBA is single-threaded, OnTime doesn't mean your code will run at that specific time, rather that when Excel isn't busy doing something else, it will queue your method for synchronous execution, like any other event handler code.

That said, your naming and access modifiers lack consistency (I guess that's what you meant with "duct-taped"?).

Observations


You're using ListObject.ListRows.Count, but you're puzzling your way into adding a new row - why aren't you using ListObject.ListRows.Add() which returns the added row?
In sPing function (awful name), I would refactor this ElseIf construct into some Private Function GetPingResult(StatusCode As Long) As String, and assign the function's return value once; also you made the default value "Alive" - if there's an error code you haven't accounted for, your function returns "Alive" which is possibly wrong.


Nitpicks


This is VBA - stick to PascalCasing for all identifiers. It will make your code read consistently with the language itself.
The default access modifier for Sub and Function is Public - thus, either specify it or leave it out, but don't do both. If the unspecified ones are supposed to be Private, be explicit about it.
The language's convention for underscores in procedure names, is only for methods that implement interface methods (e.g. event handlers). Avoid it.
Worse than Hungarian notation (strTableName As String), is inconsistent Hungarian notation, especially in the same method signature (why isn't it iCol As Integer then?). Avoid it (Hungarian notation, that is!).


I know this is an incomplete review, but I think you have enough meat here anyway.