“I am writing to you from Ireland. My grandmother was a naturalized citizen of the U.S. She was naturalized in 1951, returned home to Ireland two years later, married and immediately started a family. I am 20 and considering a move to the States, but I want to be able to go legally. Can my grandmother’s citizenship help in any way?”
Not likely. Grandparents are not permitted to pass citizenship to grandchildren, but it can go to a child, which means your relevant parent may have been a U.S. citizen at birth.
We’ll have to scroll back in time to discover if the instances when U.S. citizens can automatically pass citizenship to a foreign-born child apply in your family’s case. The requirements for doing so depend on when the child was born, as there are different guidelines in place for different periods of time.
You don’t say how old your parent is, but from the information in your letter it’s clear that (s)he was born sometime in the 1950s. The rule for passing citizenship from parent to a child born between December 24, 1952-November 13, 1986 is as follows – if only one parent was a U.S. citizen at the time of the child’s birth, the parent must have resided in the U.S. for at least 10 years, with five of these years taking place after the parent turned 14 years old.
If this holds true for your grandmother, then your parent would automatically be a U.S. citizen. No period of residency here, or any other requirement, would need to be met.
But even if your parent is a U.S. citizen, you will not be eligible to obtain an immigration benefit yourself. Though your parent would technically be able to file a petition for you as an immediate relative (unmarried children aged under 21 of U.S. citizens), the requirement of an affidavit of support for most family-based petitions would effectively render the petition because your parent is not a resident of the U.S., or any of its territories or possessions.
The affidavit of support paperwork (Form I-864) must accompany virtually every family-based immigration petition, and it must be filed primarily by the sponsoring relative. The affidavit basically promises that in the event the person seeking an immigration benefit becomes a public charge, the affidavit sponsor will be responsible for repaying the government for the cost of these services – for instance, food stamps, Medicaid, or Supplemental Security Income. (A couple of exceptions where the sponsor would not have to repay – emergency Medicaid and school lunches.)
As mentioned, the affidavit must be filed by the sponsor, who must be domiciled in the U.S. One of the several documents required to complete the I-864 is a U.S. tax return. Obviously, as your possible U.S. citizen parent is resident abroad, it would be impossible to provide a properly executed I-864, and as a result the immediate relative petition would be denied. 
Of course, if your parent is a U.S. citizen (and if this is the case, it would have to be proved through a paper trail of documents, including relevant birth certificates, etc.) there would be nothing preventing him or her from relocating to the U.S. and establishing residence here. If this were to happen then it would be possible to eventually become legal through your parent.
Clearly, though, this would be a massive upheaval for your family. But one never knows what may happen down the line, and if your parent is a U.S. citizen it’s a valuable option to have.