Rule 44.1 states that a boat may take the penalty when she may have broken one or more rules in the same incident. 

I would write facts for the two scenarios you suggest:

Boat A hit Mark 1. 
The mark bounced off of Boat A, then hit Boat A again before she passed the mark. 

or 

Boat A hit Mark 1. 
After passing mark 1, boat A hit mark 1 again while completing a one-turn penalty. 

My conclusions would be:

Boat A touched mark 1 multiple times in the same incident and broke RRS 31.  

or

Boat A touched mark 1 and broke RRS 31. 
In a separate incident, Boat A touched mark 1 and broke RRS 31 again. 

This could clearly go to an appeals committee for a definite decision, but my opinion is that these would satisfy an appeal. 

I think if a boat make multiple touches in the "same incident" then that is one penalty.

If it is touched again, in a different incident, then it is a new penality.

Edit: what John said

def : Incident - an occurrence of an action or situation that is a separate unit of experience.  Merriam-Webster

Yes, he needs to take a second penalty turn as these are separate incidents.  He was taking a penalty turn for the first incident and while doing so has broken another rule, albeit the same rule he broke in the first incident.  The mark is still relevant to his course, that is it is either the mark that ends the leg he just finished or starts the next leg of the course, so touching it breaks rule 31.

In the situation where the mark is bouncing on and off the boat as it goes by in a single pass, that is considered one incident and so rule 31 is only broken once.

For those of you reading this in the US - you should look at Appeal 65.  The USAC uses the term 'inevitable result' as the way to decide whether a situation is one or more incidents.

I'm not a judge or race officer, but a long-time sailor interested in the rules.

It sounds like two separate incidents, to me.  Bummer, though.  Get well clear of the mark before doing turns!

Agree with Chris and John.
For the future - 44.2 says take the Penalty After getting well clear of other boats - I would add that this logically means getting well clear of the Mark.  We have too many instances of competitors taking their penalty right away without looking around - and they slam into another Dinghy.  

Dana, I'm not sure that follows. A boat can take her penalty by spinning around the mark if she chooses to as long as she's well clear of other boats. But if there are other boats approaching to round then the boat taking a penalty must stay well clear of those boats.

Good point!  Nice recovery move if you can pull it off.

Yes - as John says the mark begins the leg he was sailing on so if he touches it at any time before the next mark, even after sailing well clear, he has to take another penalty.  That does feel perverse though, so I'm not surprised you got conflicting advice! 

In the USA, we have the benefit of US Sailing Appeal 65 to deal with situations like this. One of the salient portions of the Appeals Committee's decision was:
“The test of whether two occurrences were one or two incidents is whether the second occurrence was the inevitable result of the first. Times, distances, the actions of each boat and the prevailing conditions are all relevant to this test; the number of rules that may have been broken is not.” 
Therefore, in the US, a protest committee would probably find that a second circle was necessary.

Clark - Appeal 65 is actually not a benefit in my opinion.  It's there and we in the US have to follow it.  That and other appeals the USAC has decided based on Appeal 65 are quite questionable IMHO.  The notion of 'inevitable result' is really bad.  Probably a topic for another thread.  Consider two boats sailing W/L and overlapped.  They sail that way for a while getting closer slowly until W is no longer keeping clear.  They continue to sail that way for a while and then there is contact.  Is this one incident or two?  Was the contact an 'inevitable result' of the failure to keep clear?  How many turns should the windward boat do?  I'm happy to take this off-line with you and show you more problems if you email me.

John: You make a good point, but in the question described above, the decision from Appeal 65 offers clear direction.
It also occurred to me that the protest committee in Appeal 65 COULD have protested FS 112 for what the Appeals Committee called "the second incident" under rule 60.3 and disqualified her. That hardly seems equitable to me: The boat acknowledged that she had failed to keep clear of another boat and took a Two-Turn Penalty but gets tossed later in the hearing. This would seem to be "an abuse of prosecutorial discretion" but there seems to be nothing that would prevent a protest committee from taking that action.

We had a case where both boats were disqualified by the PC in what the boats, PC, and AAC all considered one incident.  The USAC decided that there were actually two incidents, arbitrarily decided which incident the protest was regarding, and reversed the DSQ for one of the boats even though that boat caused contact.  This is a case where the PC could have done what you suggest.  I can easily diagram a very typical mark rounding situation where there are 4 incidents and one boat would be required to do 5 turns and the other 1 turn to take all the penalties.  The implications are a mess.  With Appeal 65 in place, the sailors will have no idea how many incidents they are involved in, how many times to hail protest, how many protests they should file, and how many turns are required.  And anywhere up the chain of the PC, AAC, and USAC the number could be changed.

A problem I see with Appeal 65 is that still the only required action to inform of the protestee is a hail of "Protest" and (if required) displaying a red flag. So what is a protesting boat supposed to do when there are two incidents close in time to "inform the protested boat that two protests will be lodged"? 

If a boat hails "protest" twice in fairly quick succession is the other boat to assume that she's being protested for two incidents, or just that the protesting boat wants to make sure her protest was heard? What if the skipper of the protesting boat hails "protest" and then the tactician says "protest"? 

Must a second red flag be displayed? Or are you supposed to shake the one that's already displayed?

John:
Unfortunately, I have deeply imprinted upon my soul the directive that I heard at my very first US Sailing Judges Seminar in 1998.
The seminar leader, Tom Farquhar, asked, "What is the purpose of having judges at a regatta?"
There were several responses from the attendees. They included, "To hear protests", "To enforce the rules", and "To review the actions of the organizing authority, the race committee, and the competitors."
After all of the offerings, Tom said, "No. The purpose of Judges is to insure the fairness of the event."
I have always thereafter been guided by that advice.

Clark - I agree with you.  This slicing and dicing of what is an incident or will turn them off completely.  It will be like the NFL deciding whether it is a catch and fumble or an incomplete pass.  If we start to tell them well after the fact how many incidents they were involved in and then whether the PC is going to protest or not, how is that fair?  How are we serving them then?  They will just look at us like we are crazy.  We have to be guided by what they think is an incident on the water at the time and not some contrived way of parsing some words.  If anything, the definition of an incident should be broad and not narrow.

And, while not germane to the arguments, we assume your son successfully rounded the mark on the first attempt...