Does slapping a magic tattoo on an eidolon with Inscribe Magic Tattoo work? If so it might be a way to work around the item slot problem. (or... you could just slap it onto the summoner and give the eidolon all the items x_x)
It comes down to your DM in most cases as eidolons occupy a weird area. Talk to them before making plans, but in my opinion yes. They are cleaely designed to benefit from magic items due to the slot sharing mechanic and a magic tattoo is just a cooler magic item. Same goes for the permanency spell.
Probably, but there could be issues
Paizo creative director James Jacobs in a 2012 messageboard post explains that his reading of the summoner's eidolon would have a summoner's eidolon be the same eidolon each time it's summoned, the eidolon bringing back with it next time any gear it was given last time, for example. By extension, an eidolon that gets magic tattoos that were created by the feat Inscribe Magic Tattoo and then departs should still have those magic tattoos when the eidolon is summoned anew. (While Jacobs isn't an official source for rules—humbly admitting to not being a rules guy—, his rulings nonetheless carry some weight in the Pathfinder community as perhaps the game's most accessible source of semiofficial clarifications.)
[T]he summoner and his eidolon share magic item slots. For example, if the summoner is wearing a ring, his eidolon can wear no more than one ring. In case of a conflict, the items worn by the summoner remain active, and those used by the eidolon become dormant.…
And the feat Inscribe Magic Tattoo says, "Magic tattoos must be placed on a part of the body normally able to hold a magic item slot, but they do not count against or interfere with magic items worn on those slots. A single slot can only hold one magical tattoo." Taken as a whole, it's possible for a GM to rule that the summoner can't also have magic tattoos that occupy the same body slots as the summoner's eidolon's magic tattoos… and vice versa. This means that the GM could rule that inking thoroughly both the summoner and eidolon is impossible and that a summoner that herself takes the feat Inscribe Magic Tattoo won't get the mileage from the feat that she might've originally thought that she would.
Also, there are only a spare handful of magic tattoos already published, and many of them won't be particularly useful for the eidolon, like the tattoo that'll let the eidolon 1/day look like a specific orc, for instance, or the tattoo that can 1/day apply the feats Still Spell and Silent Spell to one of the eidolon's nonexistent spells. The best of the so-far published tattoos appears to be the swirling smoke tattoo… that has the not inconsiderable price of 12,500 gp.
That is, while the Inscribe Magic Tattoo feat says, "Magical tattoos follow the rules for magic item creation as though they were wondrous items, except that they can use the Craft (calligraphy, paintings, tattoos) skill," that does not mean that the feat Inscribe Magic Tattoo can be used as a substitute for the feat Craft Wondrous Item; it means, instead, that the creation of magic tattoos follows these rules. The feat Inscribe Magic Tattoo still meets only the requirements for magic items that have as a requirement the feat Inscribe Magic Tattoo and not those magic items plus magic items that have as a requirement the feat Craft Wondrous Item! In other words, unpublished tattoos that the artist wants to create can be created only with the GM's permission, and a GM may rule that new magic tattoos be deliberately different—thematically or functionally—from wondrous items to prevent the feats from overlapping.
Finally, just as an aside, while the feat Inscribe Magic Tattoo says that both the creature that possesses the feat Inscribe Magic Tattoo "and the recipient of the tattoo… must be present during the entire tattooing process," this doesn't mean, for example, that the summoner must stay awake for 13 days while her eidolon gets a swirling smoke tattoo. (That is, according to the description of the summoner's eidolon, "If the summoner is unconscious, asleep, or killed, his eidolon is immediately banished.") The summoner can interrupt the eidlolon's tattoo process—by sleeping or getting coldcocked in a bar fight or just going off with the eidolon on an adventure—then have the eidolon return to the artist so he can work on that tattoo some more… so long as the artist doesn't start creating another magic item in the meantime. (If the artist does start creating another magic item before finishing the previous, raw materials used for that previous magic tattoo are wasted as per here.)