For the record, March is the only one where we can be 100% confident that it's named after a god.

The 'Janua-' in January is the Latin for 'door, entry': the link to Janus is a retrojection. Some sources claim the Agonalia festival on 9 January was in honour of Janus, but Agonalia also took place in May and December, so there's nothing in that.

February is indeed named for the februa offerings at the Lupercalia festival. We occasionally see the Lupercalia referred to as dies februatus, 'the februated day', which is a bit of a giveaway.

For April some linguists have suggested a link to ab 'away from, off' and aperi- 'open', with a conjectural adjective *ap(e)rus supposedly meaning 'following, second' (i.e. the second month of the year, from a time when March was the first month). But it's much more likely that it comes from Etruscan afr, apru(n) 'fortune'. Fortune was also an Etruscan goddess, so this could be a second month named after a divinity. It's possible that the early Roman calendar had its own native name for the month and that it was deprecated at some point: if so, the deprecation would presumably have been in the time of the Etruscan kings.

(Edit: I'm deleting this bit about Etruscan because I haven't been able to turn up any good substantiation. It was only conjectural anyway.)

May and June come from the words for 'older' and 'younger'. It's possible that May is also connected to Maia, but June isn't directly linked to Juno. If it were named after Juno the adjectival form would have to have been Iunonius, not Iunius. It's still possible that the 'older' and 'younger' might be older and younger divinities, but there isn't a trivial link to Juno.

As for the others: the usual take on things is that March was originally the first month in the calendar, and January and February were either added, or transposed from the end of the year. In support of this there's the fact that intercalary periods were always near the end of February, specifically six days before the start of March. (And of course we still have our intercalary days in February today.)

We don't know why the Romans only had a few 'unique' names for the months, but the relationship between month names and male praenomina (personal first names) suggests that it's a very early practice. Corresponding to March, May, June, Quintilis, and Sextilis we have the common names Marcus, Maius, Quintus, and Sextus; Decimus ~ December is also common. Praenomina corresponding to June and September-November are rare or unattested. Some of the rare ones are filled in by gentilician names: Marcius, Maius, Iunius, Quinctius, Sextius, Septimius, Octavius (> constructed praenomen Octavianus), Nonius, and Decimius/Decius. These links are paralleled by some Samnite naming customs (e.g. Oscan Pomponius ~ Latin Quinctius, both from words for 'fifth').

There aren't any corresponding to January and February. That strongly supports the reading that (1) the links between month names and personal names were early, and the associations evaporated early on; and (2) January and February were added on to the start of the calendar at a later (but still early) period.

This is also why I suspect April comes from Etruscan rather than being an archaic word for 'second'. The Romans could well have had a common personal name corresponding to an earlier native name for April. That's conjectural, mind. A good candidate for that would be any personal name that exists both in praenominal and gentilician forms, like (for example) Gnaeus/Naevius.

Incidentally, don't put any stock in theories that credit the legendary king Numa for any of this. No records survived from that period, Numa probably wasn't real. It's just ancient supposition. Varro was probably right about March originally being the first month, though, and about the relationship between month names and personal names.

For the links between month names and personal names, the standard write-up is: Hans Petersen (1962), 'The numeral praenomina of the Romans', Transactions and Proceedings of the American Philological Association 93: 347-354. I also did a write-up a couple of years ago here on my own site.

Edit: spelling.