Enabling a module leads to memory exhaution. Where to look for?
A short answer for the question is to check the function names defined in the module.  If there is any conflict with the existing function names, that can give a trouble when enabling the module.  Also, it may be advisable to disable and enable the module, whenever you add a function to the module, maybe, to detect a potential trouble at the earliest opportunity.
--The following appended--
Another preventive ways: (1) Keep only hook_ functions in a *.module file, (2) for D7, name always your private functions with a little convention, for example, add the prefix of _: e.g., function _myfunction_theme().
--Appended up to here--

It turned out the function name My_Module_Name_theme() conflicts with hook_theme() in Drupal.  I was not using it as my intention, but the code of the function was left for the future development - that caused a trouble, as it was regarded as a valid hook by Drupal, whatever the programmer's intention was.  The arguments given to a call of another Drupal function in that function were not compatible with the Drupal function.  That may have lead to the memory exhaustion.

My guess is, since it is the name of the function (or maybe because it is a hook?), that caused a memory exhaustion while enabling the module rather than during run-time.  I added the function in the module when the module had been already enabled.  At that time (I mean, after I added the function), the module caused no trouble and kept functioning as I intended.  It is possibly because the function names were not registered to Drupal?  I don't know, but whichever, it would make (and indeed did make in my case) a programmer's job to track down the cause of the memory exhaustion difficult, as the error message might not help much to pin down which module, let alone which function, is causing a trouble.

Note even after I had added the "hook" function, I have once successfully disabled and enabled the module (after Drupal experienced a memory exhaustion), and the module worked fine for some time, before it eventually started to cause a memory exhaustion and then it became impossible to re-enable it.  I do not know why it was enabled successfully once and worked OK for some time.

The moral of this case is summarised at the beginning.  Added to that, to chop off all the unnecessary parts like debugging may help (as @rooby suggested in the comment of the question), because something in those unnecessary parts, or those that you think are unused, may be causing a trouble.  Frequent disabling/enabling cycles, as well as clearing the cache, can also help to identify the problem at the earliest opportunity.