It has been proposed that the alternation of *i/*e in the reduplication is maybe the result of two different reflexes of the same vowel. Beekes (1995: 227) suggests it has to do with the accent where *é is the accented form and *i the unaccented reflex. The basis for this hypothesis is Skt. dádāmi (< *de-deh₃mi) and Greek δίδωμι (< *di-deh₃mi)
Beekes proposes that the accent fount in Sanskrit is original and reflects forms where the accent isn't on the suffix, while Greek form originally represents forms where the accent was on the suffix.
As a result Beekes proposes that the paradigm of reduplicated presents looked something like this:
I don't feel that this is the most elegant explanation, as it becomes quite hard to understand why the singular has a full-grade in the root while the accent isn't on it.
Personally I think that the accent will have been the same as it was in the non-reduplicated present that is, accent on the root for the singular, and accent on the suffix in the plural. The one exception being that the reduplicated present quite clearly has *-nti for the 3pl based on Sanskrit reduplicated present 3pl suffix -ati which has a ø-grade in the root in Sanskrit. If you adhere the 'accent has to be on a full-grade' rule, the only full-grade left is the vowel *e that we find in the reduplication. I honestly have no idea how the accent ended up all the way over there.
Nevertheless, we do also find the *e in reduplication unaccented in, for example, the perfect so at least the *e/*i alternation cannot be fully explained by an accented/unaccented situation. And as it is definitely possible, at least in a later stage of PIE to have unaccented *e I would like to propose a slightly different interpretation to the accent theory.
Rather than posing the the *e is accented and the *i the unaccented form. I want to suggest that the *e is the pretonic form and the *i is the pro-pretonic form.
Pro-pretonic reflexes of vowels may sound a bit strange to some, but anyone familiar with Hebrew will be able to tell you that there is indeed an abundance of reflexes of vowels that are different in pro-pretonic position than in pre-tonic position.
take the word qāṭál `he killed' from Pre-Hebrew *qaṭál the accent is on the second syllable, and in the tonic position *a remains a. In pre-tonic position the *a is lengthened to ā (if in an open syllable), while in pro-pretonic position as can be seen in the 2pl. masc. qəṭalten the *a is reflected as ə.
Maybe this pro-pretonic explantion can give us some insight in seemingly related roots where one of the variants has a *i infix.
For example gʰrebʰ- 'to grab, dig' besides gʰreibʰ- 'id.' Which may have split into two roots due to a formation somewhat like this:
It's worth mentioning that indeed the Sanskrit root gṛbh- is found with this present formation, although LIV reconstructs *gʰrebh₂- for that reason it seems a lot nice to reconstruct one root *gʰrbʰ- with the meaning `to grab' which has a *-neh₂ present formation, which caused the root to split up into 2 roots. And the third root *gʰrebh₂- is simply an understandable mistake in LIV by assuming all *-nh₂ formations originally come from roots ending in *h₂ with an *n-infix. While this is undoubtedly the origin, it is clear that the suffix became productive especially in Indo-Iranian.
A quicky look at Pokorny yielded me a few more root pairs that may go together *ned- `to roar' *neid- 'to scold, put to shame' or *neid- `to flow, stream' (roaring of a stream).
*steb(ʰ)- `post, pillar, stump' besides *steib(ʰ)- `pole, stick; stiff'
*uek- `to bend' besides *ueik- `to curve, bend'
But for these such a nice clear paradigmatic explanation isn't as easy to find.