First- if you are a from the United States you can obtain Dual-citizenship, the United States does permit this, the question is does Honduras...
Born in Honduras: It seems if you were born in Honduras and are American (or any other nationality) you can obtain Honduran citizenship.
Born in the U.S.A. but have at least one Honduran born parent: Yes, this is information taken straight of the Honduran Embassy web page :
"How do I enroll my child born out of Honduras to the National Registry of Persons (RNP)? The parent Honduran by birth, must be filed with the nearest Consulate of Honduras and request the registration of the birth of his son. You must fill out the form that will provide the Consulate and submit the following documents: The original birth certificate of the child; A photocopy of the Act; Two photographs, and A photocopy of identity cards or passports of parents. The Consulate will issue a document that the applicant must submit to the National Registry of Persons in Honduras. The document may be submitted in person or by another person acting on behalf of the applicant. Before presenting the document to the RNP, must validate the signature of the Consul or Consular Agent, Authentic to the office of the Foreign Ministry in Honduras."Married a Honduran: Yes, you can obtain Honduran residency. The U.S.A. Embassy to Honduras is a great resource here
Honduran born Honduran: A Honduran born Honduran can keep their Honduran Citizenship and obtain United States Citizenship as well.
General Honduras Travel info:
Honduras does not require a visa to visit unless you are from on of these countries (question 4). When entering Honduras you will be granted a 90 day visa, be sure to check when they stamp you. After 90 days you can get a 30 day extension at the Direccion General de Migracion y Extranjeria (Offices in Tegucigalpa, San Pedro and La Cieba) . After a total of 120 days you will have to leave. The closest places you can go is either Costa Rica or Belize do to the Central American 4 Border Control Agreement . I have in the past spent four days out of the Honduras, and then returned with no problem but I don't know if this is legal. I hope this info is helpful, please leave a comment if you have any questions or corrections.
Info/sources http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html http://www.hondurasemb.org http://honduras.usembassy.gov http://www.rnp.hn http://www.migracion.gob.hn http://en.wikipedia.org/wiki/Central_America-4_Border_Control_Agreement