Implications in crossing the Ireland/Northern Ireland land border for an American

When someone transits via Dublin into the UK (in your gf's case to Northern Ireland), they do so on a 90 day leave-to-enter issued by the Republic of Ireland. This allows the holder to move about the Common Travel Area.

The leave-to-enter does not automatically become a six-month UK leave-to-enter by crossing a border. Regardless of where they are the person becomes an overstayer on the 91st day.

In the case you have described, your gf will not have permission to stay in the UK and will become an overstayer in two countries...

1. The UK, because her 90 leave-to-enter has expired; and
2. The Republic of Ireland, because her 90 day leave-to-enter has expired.

So when we depart from Dublin to go back to America, they will think she has been staying in ireland illegally?

Not exactly, they will think (and correctly so) that she is an illegal entrant. This is because when you return to Dublin your gf will have entered the Republic (crossing the NI land border) without a valid leave-to-enter. They will detain and serve her a removal notice and then tell the UK about it. The UK will turn around and tell Schengen and the Five Eyes about it. To make matters worse, the UK has a policy (published in their operations manual) that anyone using the CTA who has a prior removal is automatically an illegal entrant. Alternatively, she can use personal impact, see below.

Is there anyway around this problem?

There are several ways to prevent a problem...

1. Channel hop. This means to leave the CTA before the 90 days are gone and get a new leave-to-enter upon return (expensive and risky);
2. Per the comment by 'phoog' (to whom thanks), you can get married (or establish a durable relationship) and then use the relevant EU directives to obtain a family permit. Questions on this strategy should be taken up at Expats.
3. Arrive in the UK and get a full 6 month leave-to-enter (people do not like this because of the airfare);
4. Get a UK entry clearance (or the Irish counterpart) before leaving the US (cheapest, lowest risk, etc)

Options 1, 3, and 4 do not use the family formation route. Option 2 entails the lengthy wait for the permit. Option 4 is what a solicitor would tell you to do.

So much for how to prevent the problem. Given that your gf is already in the UK the options are more restricted...

1. If she is not already in breach, have her channel hop to Calais and then arrive in Dover or Liverpool or any other UK port of entry; this is risky because she will be up against V 4.2 (b) of the rules (successive visits) during her landing interview.
2. If she is already in breach or a channel hop is not an option, take the full hit on the chin and claim to be an accidental overstayer/illegal entrant.

The outcome for both of these options is determined "solely" by personal impact and articulation skills. If those skills are up to the task, then she can get a verbal reprimand and walk away with no negative immigration history. People do this all the time.

There is also the possibility that her breach will not get picked up. But it will all come out in the wash if she goes for a settlement visa like fiance/spouse or a family permit. Pay for her departure flight with your credit card, have her avoid using her credit card whilst in NI.

Miscellaneous Notes and Other Ramblings

People often ask if they can simply approach a UK Immigration Officer from landside and get their passport stamped with a UK leave-to-enter. IO's will not do it, it's a sackable offense. The only way to get a UK leave-to-enter is by arriving in the UK directly from outside the Common Travel Area. Source: Immigration Act 1971

"Leave-to-enter" is the official term for what US citizens get when they arrive in any of the Common Travel Area countries. It takes the form of a rectangular passport stamp and functions exactly like a visa. Same act as above, but referred to as an "entry certificate", the two terms are synonymous.

The US Embassy in London is (or was) running an advisory on precisely this topic warning travellers not to become an accidental overstayer.

Students, fiances, spouses, and work permit holders do not have the problem because they already hold a UK entry clearance, so keep this in mind when reading anecdotal stories on the net (also, spouses and fiances who transit in Dublin have a completely different can of worms to sort out).