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Yeah, for a single American citizen parent there are two possibilities:

According to 8 USC Section 1401

d) Parent lived in the U.S.A. (or outlying possessions) for a period of at least one continuous year prior to birth of child, if the other parent is a U.S. National but not a citizen. 8 U.S.C. § 1408 stipulates that a person whose only connection to the U.S. is through birth in an outlying possession (which as of 2005 is limited to American Samoa and Swains Island), or through descent from a person so born acquires U.S. nationality but not U.S. citizenship.

g) Parent lived at least a total of 10 years within the U.S.A. (or outlying possession), and at least 5 years total after the age of FOURTEEN.

Pertainent ammendments to this section of the code:

1986 - Subsec. (g). Pub. L. 99-653 substituted "five years, at least two" for "ten years, at least five".

This ammendment is not applicable in a foreign birth situation to a U.S. citizen parent prior to the date of the ammendment, in that Congress is prohibited from passing any ex post facto laws. The point is that that is the ONLY ammendment to that part of the statute; the age FOURTEEN as been inviolate since the statute's inception.

Given that Obama's father was NOT a U.S. National, 8 USC Section 1401(d) is not applicable. Therefor, Obama's mother MUST be no younger than 19 at the time of NoBama's birth, AND she must've resided in the U.S.A. (or its outlying possessions) for 5 continuous years prior to his birth. This of course is ONLY relevent if NoBama wasn't born within the U.S.A (or its outlying possessions). And the older than 19 NoBama's Mamma when he was born, the longer the time she could've been outside of the country. So if she was 20 when NoBama was born, she could've been outside of the country for a total of ONE YEAR after she turned 14.

IF he was born within the U.S.A. (or outlying possessions), then he is entitled to American citizenship (not naturalization) as matter of Law jus soli. As a practical matter, citizenship is conferred to the child in accordance to the Father's nationality and / or citizenship as a matter of jus sanguinis. Nevertheless, and that notwithstanding, jus sanguinis does not abrogate inherent rights of jus soli.

I believe that it is almost universally accepted that one can reclaim one's "birthright" at any time. However, there are mitigating factors in which jus soli would irrevocably be relinquished; such matters would be deliberate acts and most likely after the age of majority.

With respect to dual-citizenship its complicated. The U.S. Immigration Support web-site states that dual-citizenship is recognized by the U.S. government. It is stated that

The U.S. government allows dual citizenship. United States law recognizes U.S. Dual Citizenship, but the U.S. government does not encourage it is as a matter of policy due to the problems that may arise from it. It is important to understand that a foreign citizen does NOT lose his or her citizenship when becoming a U.S. citizen. An individual that becomes a U.S. citizen through naturalization may keep his or her original citizenship. However, as some countries do not recognize dual citizenship, it is important to consider it carefully before applying for U.S. citizenship.
Dual citizenship is a complex issue and it is important that you understand that there also obligations and not only benefits that comes with being a dual citizen. Being a citizen of two countries means that you need to obey the laws of both countries, including paying taxes and serving in the military (if required by any of the countries)...

I've always percieved citizenship to be that of whatever country you live in and pay taxes to. However, I can identify with cultural ties that preclude one from severing citizenship ties to the "old country" of one's heritage (even if its "vestiginal" in that no obligation or onus is placed by the country of one's heritage.

The biggest issue with regards to citizenship pertains to travel. International travel requires passports and quite possibley either exit or entry visas (or both). Having dual citizenship allows one to enter on a foreign passport and entry visa is not required by the destination country. However, the reverse may not be so clear.

The destination country may require an exit visa to leave the country on a "foreign" passport, i.e., American. Returning to the "home" country with the "home" passport from a foreign destination will not contain an exit stamp of the "foreign" country on the "home" passport. That could require an entry visa to the "home" country. Its not quite "Catch-22", but it can get sticky; you either don't get out easily of the "foreign" country, or you may not get in to the "home" country so easily. Its always been my thought that the point of having two passports was so that one could get out of hairy situations when needed. Nevertheless, the issue of exit / entry visa is a potential issue; how would one explain to that particular interested "gatekeeper" why such visa isn't in the passport currently being used? Dual / multiple passport travel is definitely not for the faint hearted, and especially if one doesn't know what they're doing.

I heard tell:

On the Vietnamese-Lao border I met a very poor guy who had left Lao using his NZ passport, and trying to enter Vietnam om his English one. The Vietnamese wouldn't let him in and didn't acknowledge that it is possible to have two passports. In fact they thought he was a criminal with two passports. The Lao would not let him in anymore since he did not have a valid visa anymore. I hope he is not still sitting there...