The short answer is yes, a foreign national who has more than one close relative in the U.S. with U.S. citizenship or permanent residence status need not choose only one of them to start the immigration process (leading eventually to a green card) for him or her. Any or all of the person’s U.S. relatives can serve as “petitioner.”
Of course, only a limited number of family relationships potentially qualify a foreign-born person to receive a green card to immigrate to the United States. These are described in Green Card Through a U.S. Family Member: Who Qualifies.
Note on terminology: The U.S. citizen or lawful permanent resident family member is called the “petitioner” because he or she must file a so-called “visa petition” on the intending immigrant’s or “beneficiary’s” behalf (using Form I-130, issued by U.S. Citizenship and Immigration Services or USCIS). Some people call the petitioner a “sponsor.” However, in legal terminology, “sponsor” actually refers to the person providing financial sponsorship to the immigrant, which many people in the U.S. might be able to do (although the petitioner must, as part of the immigration process, be the first and primary financial sponsor).
Different family relationships may lead to a U.S. green card at different rates of speed. The spouse, parent, or unmarried child of a U.S. citizen, for example, is considered an “immediate relative.” No annual limits apply in the immediate relative category, so the beneficiary’s application for a green card will go just as quickly as the U.S. government’s bureaucracy and application requirements allow.
All other family relationships that qualify a foreign national to immigrate, however, fall into the “preference” categories, in which the law limits the numbers of immigrant visas or green cards given out each year. Due to high demand for U.S. green cards, the upper limit is invariably reached, leading to waits of many years in each category. The worst category is sibling (brother or sister) of a U.S. citizen, in which the wait for an available visa is, on average, at least ten years – and often many more.
Therefore, it’s worth figuring out which family relationship will get the foreign national a green card the fastest, and make sure that the appropriate petitioner files (if willing) an I-130 on the immigrant’s behalf. You can do this by looking at the monthly Visa Bulletin published by the U.S. State Department, which lists both the preference categories and the “priority dates” (I-130 submission dates) of people who have become eligible for visas in that month.
The farther back in time the listed priority dates, the longer the wait. For example, by looking at the Visa Bulletin, you might notice that a married person with U.S. citizen parents as well as siblings will get a green card slightly faster through the parents, because category F3 is moving about a year ahead of category F4 (for applicants from most countries, that is; people from Mexico and the Philippines wait much longer than others, due to high demand from those countries).
If one of the potential petitioners is the intending immigrant’s immediate relative, so that the entire application process will go fairly quickly, you could simply file that one application, and assume all will go well. (You’ll find out fairly soon, in any case.)
In any other situation, however, it can be well worth having more than one petitioner file an I-130 on the immigrant’s behalf. Why? Because circumstances can change with the passage of time. A petitioner might die or lose interest in helping the immigrant.
For example, let’s say that U.S. green card holder Eileen marries Ivan, a citizen of Bulgaria. Because Eileen is a permanent resident, Ivan will be in category 2A of the visa preference system, facing a wait of, on average, at least two years before a visa becomes available to him. After living apart for nearly two years, however, Ivan in Bulgaria and Eileen in the U.S., their marriage falls apart. Thus Ivan is no longer eligible for an immigrant visa through Eileen. Fortunately for Ivan, he also has a brother who is a U.S. citizen, and that brother files a visa petition for him. Siblings of U.S. citizens are in the 4th preference category of visas, in which Ivan is now looking at an 11-year wait until a green card becomes available to him. If only Ivan’s brother had filed the visa petition for him a couple of years ago, he could be looking at an already much shorter wait.
What if the petitioner dies before the immigrant gets the green card? Some possible remedies exist in such cases, including something called "humanitarian reinstatement" and a law assisting surviving spouses of U.S. citizens described in Can Immigrant Still Get Green Card After U.S. Petitioner/Sponsor's Death? Nevertheless, such remedies may, depending on the facts of the situation, not work in all cases.
The bottom line: You have nothing to lose except the filing fee by having more than one U.S. petitioner file an I-130 on behalf of an intending immigrant, and doing so may save a great deal of time and heartache.