It's probably best for an elemental ally druid who wants to cast summon eidolon to stick with wands and scrolls
There are a handful of other ways for a druid with the archetype elemental ally to access the spell summon eidolon or a similar effect—for example, a druid could have a summoner cast the spell summon eidolon into a ring of spell storing or an ioun stone (vibrant purple prism) or a generous GM could allow an elemental ally druid to benefit from a robe of gates—, but these tend to be more resource intensive than a few ranks in the skill Use Magic Device, some trinkets that increase the druid's Use Magic Device skill modifier, and a wand, a staff, or scrolls of the spell summon eidolon. There just doesn't seem to be any printed way to put onto the druid spell list the spell summon eidolon.
I hesitate to suggest it because it meets with resistance and disdain from players and GMs—and also because it's heavily dependent upon GM generosity and subject to strict GM oversight—, but in this case it seems like the way to go is with Independent Research: "A divine spellcaster can also research a [new] spell independently, much as an arcane spellcaster can." That said, this GM wouldn't have a problem with an elemental ally druid researching an original spell that's simply a druidic version of summon eidolon, but ask yours.
(There are many ways to add spells to spell lists in dnd-3.5e, with which Pathfinder claims it is fully compatible, but I suspect bringing in material from another game is beyond this question's scope.)
Adding effects to a summoned eidolon because of having used the spell summon eidolon
If a caster casts the spell summon eidolon, consensus appears to be that the resultant eidolon benefits from any effects that affect appropriate creatures brought forth by the caster's summon spell. That is, an elemental ally druid that possesses the feat Augment Summoning who casts the spell summon eidolon applies the feat's benefit to the summoned eidolon, and, similarly, the effect of the lesser rod of giant summoning can also be applied when casting summon eidolon. (An eidolon summoned by the ritual instead of the spell can't benefit, though.)
(This reader's original thought was that because the spell summon eidolon specifically summons the eidolon "as if you had summoned your eidolon normally, except that it only remains with you for the duration of this spell," that the eidolon would not benefit from more general effects like the feat and the rod, the spell's specificity overriding the general rules. But, as I could find absolutely no one taking this position, I won't take it either. I leave this remnant here as a possible but likely flawed reading because this reader had that notion and somebody else might, too.)
However, the feat Versatile Summon Nature's Ally can't affect an eidolon brought forth by the spell summon eidolon, the spell summon eidolon not creating an effect like the spell summon nature's ally.
Keep in mind, though, that the Pathfinder Core Rulebook FAQ includes this exchange:
Does using a potion, scroll, staff, or wand count as "casting a spell" for purposes of feats and special abilities like Augment Summoning, Spell Focus, an evoker's ability to do extra damage with evocation spells, bloodline abilities, and so on?
No. Unless they specifically state otherwise, feats and abilities that modify spells you cast only affect actual spellcasting, not using magic items that emulate spellcasting or work like spellcasting.
Thus, for example, when a wand of summon eidolon is used to create the effect of the summon eidolon spell, the wand's user's Augment Summoning feat does not apply. Whether the her casting language of the lesser rod of giant summoning applies to the effect of the wand of summon eidolon is a question for the GM.