From Old English ēam ‘uncle, esp. maternal’ and Old High German ōheim ‘maternal uncle’ we can reconstruct Proto-West Germanic *auhaimaz ‘maternal uncle’. (PWGmc *au becomes OE ēa, and PWGmc *h is lost in voiced environments in OE with contraction of resulting vowel sequences, while in OHG PWGmc *au and *ai become ei and ou respectively in most positions, but *au becomes ō before dental consonants, r and h.) This word looks like a compound consisting of a dependent noun stem *aw- and a head noun *haimaz.
The head noun is readily identifiable as the PGmc word *haimaz ‘home’ (cf. Goth. haims ‘village’, ON heimr ‘home, world’, OE hām, OHG heim; the ‘village’ sense is also attested in place names in the NWGmc languages), which does not (AFAIK) have direct cognates in other IE languages, but it can be derived from a formation *ḱóymos ‘resting place’ from the verbal root *ḱey- ‘lie down’ (cf. Skt śete ‘(s)he is lying down’, Gk keîtai ‘(s)he is lying down’). As for the dependent noun stem, it appears to be the same one seen in Goth. awō ‘grandmother’; presumably its meaning was ‘grandparent’, or ‘grandfather’ given that non-gender-specific IE stems used to form kinship terminology tend to refer to males by default. The stem would have been *awa- in PGmc, which became *aw- in PWGmc after the loss of word-final low vowels that occured in the WGmc languages. There are cognates outside of Gmc too; PGmc *awa- would be the regular outcome of PIE *h₂áwh₂o- ‘grandfather’, which is widely attested, via reflexes of the noun *h₂áwh₂os formed directly from this stem (cf. Hitt. ḫuḫḫas, Toch. B āwe, Arm. hav, Lat. avus ‘grandfather, ancestor’), via reflexes of the adjective *h₂áwh₂yos (cf. OIr. aue ‘grandson, descendant’, Serbo-Croat. ȕjāk ‘maternal uncle’ [with a dimunitive suffix -ak]), and via n-stem reflexes of some form or other (cf. Lith. avýnas ‘maternal uncle’, Lat. avunculus ‘maternal uncle’ [with a dimunitive suffix -culus], OIr. amnair ‘maternal uncle’ [with the ending of athair ‘father’]; also Goth. awō itself, which has the gen. sg. awōns). As we can see from these reflexes, it wasn’t uncommon for derivatives of *h₂áwh₂o- to come to mean ‘maternal uncle’. This is probably due to the fact that in strongly patrilineal societies (and it is thought that PIE society was strongly patrilineal), there is a tendency for kinship terms to conflate members of different generations who are relatives on the mother’s side. Note also that the parallel conflation of nephews with grandchildren is widely attested in IE, and from Gothic and Slavic there is evidence that the conflation was originally with sororal nephews, specifically.
Now, something is very weird about this compound noun (PGmc *awahaimaz, PWGmc *auhaimaz). The two constituent nouns in a compound noun don’t usually have an equal relationship. Generally, one of them can be identified as the head noun, and the other can be identified as the dependent noun; the compound is used to refer to things that can be described as the head noun, but have some more specific attribute, not implied by the head noun itself, described (or often only vaguely alluded to) by the dependent noun. For example, in English, a blackbird is a bird which is black, and a football is a ball which you kick with your feet. Generally, in English and the other Germanic languages, and in PIE itself, compounds like this (which are called determinative compounds) are head-final, that is, the head noun is the second noun in the compound, and the first noun is dependent. But it seems unlikely that a compound whose head was *haimaz, and which therefore referred to a kind of home, would have ended up meaning ‘maternal uncle’. Instead, it seems that the head of *awahaimaz has to be *awa-, the first noun in the compound—unlike every other PGmc compound I’m aware of.
Furthermore, even if we assume *awahaimaz was a head-initial compound meaning ‘home-uncle’ or something like that, it’s still unclear what the dependent noun *haimaz is meant to indicate here. In a patrilocal society, wives live in the home owned by their husband’s family after marriage. Hence, the members of one’s extended family who live in the same home may include one’s paternal uncles, but it does not include one’s maternal uncles (because one’s mother has come from a different home, where she lived before getting married). It would therefore make more sense for paternal uncles to be referred to as ‘home-uncles’, rather than maternal uncles. We know that the Gmc peoples were patrilocal in historical times and we are reasonably sure that the speakers of PIE were patrilocal. It’s possible that PGmc speakers passed through a temporary matrilocal stage, in which case the compound would make more sense, perhaps. But if it was common for people to live along with their paternal or maternal uncles along with their parents, it is quite likely that these uncles would have been referred to as ‘father’ or by some transparent derivative of that term; after all, they would have served a similar role, and such conflations are known to be common in strongly unilineal (that is, either patrilineal or matrilineal) societies. So I don’t think this is a totally convincing explanation; and it doesn’t explain the fact that the compound is head-initial in any case.
Anyway, I don’t have a solution to this mystery. I just thought it was worth pointing out.