minhag to visit cemetery

Rama at the end of OC 581 quotes the minhag of visiting the cemetery before R"H:

ויש מקומות נוהגין לילך על הקברות ולהרבות שם בתחינות

My mother treats this minhag like a yei'hareg v'al yaavor.  Once the calendar hits Elul, she already starts planning when to go to each cemetery that relatives are buried in.

Aliba d'hilchisa, there does not really seem to be any source for this.  The place where we DO find a source for visiting the cemetery is in hil taanis (based on Taanis 16), which is the source for Rama in OC 569 by hil tisha b'av (see here ):

והולכין על הקברות מיד שהולכים מבית הכנסת

What I think may be going on in hil Rosh Hashana is if you look back earlier in that siman you have the din of fasting on erev R"H in seif 2:

נוהגין להתענו' ערב ראש השנה

The SA then follows that by telling us that there is no tachnun on erev R"H (seif 3), and that a person should get a haircut and wash his clothes on erev R"H (seif 4) -- everything is about erev R"H.  The Rama'a quotation of the minhag to visit the cemetery is in that context -- halachos of erev R"H.  Perhaps once the SA lays the groundwork of there being a minhag to fast on erev, the Rama extends the din of visiting the cemetery from hil taanis to this day of fasting as well.  

That doesn't help explain the idea of going during Elul, but I think it fits the words.