I've changed my mind a number of times about which of these variants works best:

1. 
   

   the second most northerly coffee shop in Seattle

   
   




2. 
   

   the second-most-northerly coffee shop in Seattle

   
   




3. 
   

   the second-most northerly coffee shop in Seattle

   
   




4. 
   

   the second most-northerly coffee shop in Seattle

   
   

I've tried to do a little research, but I'm not even sure what to call something like this, so I haven't had much luck.

Any guidance on what is considered most clear/correct would be much appreciated!

A hyphen should be used to force the parsing "(second most) northerly" rather than "second (most northerly)". Thus, it should be "second-most northerly".

I'd be inclined to write this as "the second most northerly coffee shop in Seattle", with no hyphens. But hyphenation is far from a definite area of English punctuation. I don't think any of the options that you list is unclear or ambiguous in practice.

Google Ngram Viewer data

While I'm not entirely sure of the accuracy of the Google Ngram Viewer with hyphenated phrases, it seems to suggest that "the second most common" is much more frequent than "the second-most common":

(The double-hyphenated spelling "the second-most-common" was so infrequent that it did not show up at all on the Ngram Viewer chart.)

CMOS rules that seem applicable

The Chicago Manual of Style (17th ed) gives somewhat complicated rules for hyphenating phrases like this.

It recommends using a hyphen between a ordinal number and a superlative

In the original version of this answer, I missed the fact that there is actually a special section in the CMOS hyphenation table for "number, ordinal, + superlative". This says to use a hyphen after words like "second" when they come before an attributive superlative adjective. The examples are

a second-best decision

third-largest town

fourth-to-last contestant

he arrived fourth to last

Unfortunately, none of these examples show how to hyphenate a phrase that uses most rather than -est to form a superlative adjective. The example of fourth-to-last shows hyphenation between all words in the phrase, not just after the word fourth.

The CMOS's recommendation here seems a bit unusual to me, actually. A number of dictionaries mention this use of second and give examples without hyphenation:

• 
  

  OED second, adj. and n.² 2b : "With following superlative: Having only one superior in the specified attribute."

  
  
  

  
  

  1979 Nature 15 Feb. 561/2 Secernosaurus is the second most primitive hadrosaur known.

  
  

  
  
  

  (the OED entry does give one example quotation that uses hyphenation, "1977 Word 28 104 The second-youngest of the fluent speakers.")

  
  



• MW ³second adverb 2 : "before all others with one exception · the nation's second largest city · They are my second favorite band."




• AHD
  second² adv. 2 : "But for one other; save one: the second highest peak."

So I don't think it's universally regarded as unacceptable to use a space rather than a hyphen after second in this context. The preference for a hyphen between second and an immediately following attributive superlative adjective seems to be something that is particular to the Chicago Manual of Style (I'm not sure what other style guides say about this topic).

But CMOS also recommends not using a hyphen with most, or with an adverb that is modified by another adverb

CMOS 17 says

certain compounds, including those with more, most, less, least, and very, can usually be left open unless ambiguity threatens.

For example, it recommends writing "the most efficient method" without a hyphen.

What it means by "unless ambiguity threatens" is indicated with the following pair of examples:

the most skilled workers (most in number) but

the most-skilled workers (most in skill)

So my interpretation is that, according to CMOS 17, it might be acceptable to write something like "the most-northerly coffee shops" to avoid the alleged ambiguity of "the most northerly coffee shops" (e.g. a sentence like "This street has the most northerly coffee shops in Seattle" is theoretically ambiguous, although not really in practice because "northerly" isn't an adjective that typically can apply to a bunch of different shops).

I can't find support in CMOS for using a hyphen before but not after most. It says

When the adverb rather than the compound as a whole is modified by another adverb, the entire expression is open.

The relevant pair of examples is "a much-needed addition" vs. "a very much needed addition". (It's a bit unfortunate that this uses the adverb very). Based on this, it seems to me that if second is viewed as an adverb modifying most, then you should write the whole thing with spaces according to CMOS.

Now, it's not clear to me that this is the actual function of second in this phrase. The AHD does have an entry for second as an adverb, but the OED does not, and lists this use of second under the entry for the adjective.
The fact that we can say things like "the second highest mountain", where the equivalent of the word most is expressed by a suffix -est on the adjective, suggests to me that second in this context is not used as a modifier of the adverb most (since that adverb is not even present in "the second highest mountain"). On the other hand, I guess some people can say things like "the most and second most important reason", which might indicate that "second most" is treated as a phrase (I'm not sure).

Two hyphens: recommended by "Daily Writing Tips"

Although the use of two hyphens seems to be uncommon, as I mentioned in the Google Ngram Viewer section, it has been recommended by at least one person. The Daily Writing Tips article "5 Examples of Insufficient Hyphenation", by Mark Nichol, criticizes the punctuation of “The adviser some call the world’s second-most powerful man prefers to work behind the scenes,” saying

Second-most is a nonsensical modification of “powerful man.” Powerful is part of the ranking, so it should be part of the phrasal adjective: “The adviser some call the world’s second-most-powerful man prefers to work behind the scenes.”

Surely the primary variable here is the 'northerliness' of the coffee shops' locations. Consider the following:

'The Coffee Cup' is a coffee shop in Seattle.

'The Barista Bar' is another coffee shop in Seattle.
It is further North than 'The Coffee Cup'.

'Luscious Lattes' is another coffee shop in Seattle.
It is further North than 'The Barista Bar'.

The simplest way to make clear their relative northerliness is to say:

Luscious Lattes is the most northerly coffee shop in Seattle.

The Barista Bar is the second most-northerly coffee shop in Seattle.

The Coffee Cup is the third most-northerly coffee shop in Seattle.

etc.