At the time of the conversation there are only two options introduced in scripture, 'elohiym and Yĕhovah 'elohiym. 'elohiym is used exclusively from Genesis 1:1 until Genesis 2:4 in reference to God. It is only after the first Sabbath that the use of Yĕhovah 'elohiym is recorded by Moses. The transition scripturally introduces the proper name of God, Yĕhovah which is attached to 'elohiym. The proper name of God translates to “The Existing One”. This can be seen better with the use of Young’s Literal Translation which was written as a companion to the King James Version for better academic or literal use. This also reveals the King James use of “LORD” in expressing Jehovah in English usage.
The Hebrew tradition of not pronouncing the proper name of God, due to the fact of its great holiness may factor into the use of 'elohiym in the Serpent conversation. This being the first recorded conversation in which God is spoken, means there is no reference preceding this conversation to place it in context of being proper or improper.
Eve is speaking the next two times the name of God is spoken in scripture.
Every reference to God in Chapter Four other than Eve’s statement in verse 25 uses Yĕhovah without attaching 'elohiym. The post sin relationship of Adam and Eve to God may be indicated in this usage. Moses returns to the primary use of ‘elohiym in Chapter Five. The first birth is a declaration of joy and blessing from God. The second birth is barely recorded and the third is more a statement of fact than announcement. So we see Yĕhovah in praise and 'elohiym in a statement of fact. But this cannot change the understanding of the pre-sin mindset of Eve.
The next scripturally recorded quote in which God is spoken is in Genesis 9:25-27. This gives us little assistance in that Noah uses both terms in the same statement. Although, you could argue that Noah first uses the literal phrase, “Blessed of Jehovah my God”  in this post flood period to declare his relationship to God, to his children. Noah then returns to the same practice as the first recorded conversation using 'elohiym.
Eve would not have had any reference to sin or a rejection of God verbally or by experience during the Serpent Conversation. This is not a question of intelligence, but rather a reference to her place in the timeline of creation. Therefore, there would be no reason for her to declare allegiance because she only knew the true God. (Yĕhovah 'elohiym) The conversation precedes sin; no one had ever failed to walk with God as in the latter days of Noah.
I would conclude that it would be difficult to present any argument of motive in Eve’s selection of 'elohiym in the Serpent Conversation. It would be much easier to argue that Yĕhovah is assumed because there is no alternative in the timeline of creation. Applying later scriptural truths would violate exegetical practice either from Eve in Chapter Four or Noah in Chapter Nine.
 Blue Letter Bible King James Version with Strong’s numbering of key words. http://www.blueletterbible.org/Bible.cfm?b=Gen&c=3&v=1&t=KJV&sstr=1#
 Blue Letter Bible Young’s Literal Translation (Strong’s number not available in this version) http://www.blueletterbible.org/Bible.cfm?b=Gen&c=3&t=YLT
 Blue Letter Bible Young’s Literal Translation http://www.blueletterbible.org/Bible.cfm?b=Gen&c=9&v=1&t=YLT
 Blue Letter Bible King James Version with Strong’s numbering of key words. http://www.blueletterbible.org/Bible.cfm?b=Gen&c=9&v=1&t=KJV