Hudson 1.0 has a subroutine that states "the Runner cannot access more than 1 card during this run."

With respect to Archives, the rulebook states as follows:

Archives: The Runner accesses all cards in Archives and any upgrades in its root. The Runner turns all cards faceup when accessing them, and does not need to keep them in order. The Runner steals all agendas in Archives and cannot trash cards that are already in Archives. After accessing Archives, all cards in Archives return to Archives faceup

The FAQ also has this to say about Archives:

Accessing Archives: When the Runner makes a successful run on Archives, he turns all cards faceup in Archives before accessing them. Then he accesses and resolves individual cards one by one, in any order he wants.

This leads to a couple of potential interpretations of Hudson 1.0's interaction with Archives.

1) Hudson 1.0 will protect the integrity of cards in Archives with the exception of the 1 card that the runner chooses to access. The accessed card returns to Archives faceup as normal, or is stolen if it is an agenda. The runner does not turn over any other facedown cards in Archives.

2) Upon a successful run on Archives, all cards in Archives are turned faceup, then the Runner chooses one to access, scoring an agenda if one is accessed.

3) Although the Runner only accesses one card, the Runner still steals all agendas in Archives, and cards not accessed are nevertheless returned to Archives faceup.

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As written, the rules specify three separate effects that occur upon a successful run in Archives:

1. Runner accesses all cards
   1. Accessed cards are turned faceup and need not be kept in order
2. Runner steals all agendas in Archives
3. After accessing cards, all cards in Archives are returned faceup.

The FAQ adds that all cards are turned faceup prior to access, and clarifies the order in which the Runner accesses them.

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Personally, I feel that the first interpretation is more true to the intent behind the card, and makes Hudson 1.0 much more useful. However, the rules as written allow for ambiguity in interpretation.

The second interpretation is a result of the FAQ's modification of the core rules - from "cards are turned faceup when accessing" to "cards are turned faceup before accessing." There is some ambiguity about how this interacts with limitations on accessing. Either A) only cards that would be accessed are turned faceup, or B) all cards are turned faceup, then access occurs.

If all cards are turned faceup prior to access (whether they may be accessed or not), that diminishes Hudson 1.0's value on Archives considerably. Although the Runner may only access one card, he may look at all cards in Archives before choosing which one to access and can steal an agenda if one is present.

The third interpretation is possible because the rules as written don't make it clear that accessing cards is a necessary condition to the other effects happening. The third effect, particularly, seems to draw an implicit distinction between accessing cards and the rest of the cards in Archives. This interpretation would make Hudson 1.0 useless on Archives because all agendas would be stolen regardless of whether they are accessed or not.

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Potential fixes:

In order to fix this situation, FFG might consider the following language as an errata to the Archives entry:

Archives: The Runner accesses all cards in Archives and any upgrades in its root. Accessed cards are turned faceup, then the Runner resolves those cards in any order he chooses. Agendas accessed from Archives are stolen. All other accessed cards are returned to Archives faceup. Cards in Archives do not need to be kept in order. Cards already in Archives may not be trashed.

This wording would eliminate the ambiguity, although it does change "all cards in Archives are turned faceup prior to access" to "all accessed cards are turned faceup prior to resolution."

I read it as #2.

My understanding is this: as the FAQ document provides clarification on the rules in the rule book, any ambiguity should be resolved with accordance to the FAQ. Thus, before accesssing any cards in Archives, the runner turns all cards face-up (regardless of how many cards are accessed). If under the restriction of a Hudson subroutine, the runner is then limited to accessing one card (of their choice), either in Archives or in its root. If it is an agenda, it is stolen, otherwise it is returned to Archives (or to the root if it is an upgrade that the runner didn't trash).

The runner cannot steal an agenda that he is not accessing.

Lukas has previously ruled that all cards are turned up when the run is successful. This even happens if the Corp wins the trace with Ash 2X3ZB9CY (and might even happen before that trace; I don't recall).

Based on that, there's no reason that the runner wouldn't still flip cards faceup if Hudson has resolved. It just means that they can only choose to access and resolve one of the (now all) faceup cards.

So interpretation number 2 would be the most consistent with prior rulings.

Yep, #2, no ambiguity based on the rules as written. Turning cards faceup is a separate effect from accessing, it happens automatically, then the other effects kick in. The runner chooses one card to access, then Hudson says "okay, you're done."