Short answer. Yes, if there is no doubt you are the one meant. A will should name an heir by name and surname, but even an omitted or incomplete name does not defeat the gift when the heir is designated so clearly that no doubt remains about who was instituted. Certainty of identity, not a full name, is what the law requires.
What the law says
Even though the testator may have omitted the name of the heir, should he designate him in such manner that there can be no doubt as to who has been instituted, the institution shall be valid.
Civil Code, Article 843 — Naming the Heir. Read the full provision →
Identity matters more than the full name
Article 843 asks the testator to designate the heir by his name and surname, but it does not make a perfectly written name the price of a valid gift. The same article provides that even though the testator may have omitted the name of the heir, should he designate him in such manner that there can be no doubt as to who has been instituted, the institution shall be valid. So a will that calls you "my eldest son", or gives a first name only, or a nickname, can still institute you, provided the words leave no real doubt that you are the person meant. The object is to identify the heir, and identification can be achieved without a complete legal name.
The problem the article solves
The purpose of naming an heir is certainty, ensuring the estate goes to the person the testator actually had in mind, not to a stranger who happens to share a name. The first sentence of the article addresses the opposite risk: where two people bear the same name, the testator is to add some circumstance by which the instituted heir may be known. Whether the description is too skimpy or too crowded, the test is the same, can the intended heir be pinned down with confidence from the will and the surrounding facts? If yes, the designation holds.
How doubt is resolved
When the words alone do not settle identity, the surrounding circumstances are brought in, the testator's family, who his children were, the relationships and situation existing when he made the will. This is the ordinary work of clearing up an ambiguity in identification, and it draws on the same kind of extrinsic evidence used to resolve an uncertain description, short of the testator's own reported statements of intention. The question is always factual: does the record leave any genuine doubt that you are the heir designated? A shortened name that everyone understood to mean you does not.
What to be ready to show
If your inheritance is questioned because the will did not spell out your full name, the ground to prepare is proof of identity. Assemble what ties the description used to you, your relationship to the testator, documents showing the name or nickname he habitually used for you, and the absence of any other person the words could fit. Where the designation genuinely points to no one but you, the institution is valid and the gift stands. These questions are threshed out at probate, so the identifying evidence is what carries the point.