For 30 years I have used the last name of a person to identify that the product I produce was invented and promoted by said person, now deceased for some 25 years.
The essence of a trademark is that it shows that all goods bearing that mark come from a common source. In other words, it serves as a brand. So it's important to understand what you mean by "to identify that the product I produce was invented and promoted by said person". Merely identifying the inventor of a product is not a trademark function.
If someone sells, e.g., "Smith's widgets", the name "Smith" is serving as a trademark. (Granted, being a person's name, "Smith's" counts as descriptive and so a business would need to show secondary meaning to enforce it, but it would nevertheless be a mark which could become quite solid over time. Decades of use are certainly long enough in terms of time to establish secondary meaning.) On the other hand, if someone sells Wide-world widgets and mentions that they were invented by Smith....that doesn't sound so much like a trademark use of the name "Smith."
This person's estate gave verbal permission for me to do this, and never tried to stop me from using the name to identify my product.
Technical but important point: The "estate" is a fictitious legal entity that is created by the court to help distribute the decedent's assets and pay off the debts. An "estate" cannot give verbal permission. The executor or administrator of an estate might give permission, but would probably not have authority to do this. Perhaps what you mean is that one or more heirs gave verbal permission?
Now the estate is trying to disallow use of the name by me.
Again, I presume you are speaking of one or more heirs. (It would be highly unusual for an estate to remain open all this time.)
Is it correct that ... I can claim ownership of the name in question, and register it as a trademark?
No one "owns" a "name". What you might have is common law rights in use of the name as a trademark for your kind of product.
Generally speaking, if A has a right to prevent B from doing X, but watches B do X for a long time without complaining, the law is unsympathetic to A if he suddenly decides he wants to enforce those rights -- especially when B has made substantial investments in reliance on A's sitting on his thumbs and not doing anything. So even if A didn't give permission, verbal or otherwise, if A was aware of B's actions, A can't really complain now. Equitable defenses of "laches" (pr. lach-eez) and/or "detrimental reliance" come into play to prevent A taking unfair advantage of B this way. This is particularly true in trademark law, which comes out of a tradition of equity, an area of jurisprudence that looks to "fairness". (Exception to this would be if A actually GAVE permission, but said it was OK only so long as B continued to do certain things or did not do others. That would present a different scenario.)
Also, note that if the name at issue is the name of someone famous, there may be publicity issues subject to other rules than trademark law which would come into play. Still, one would think the same equitable defenses would apply in that context.
But assuming your use of the name has been consistent and open over time, and so well known that the heirs would be aware of it, the main issue I see here is one of whether or not your use of the name has constituted a "trademark" use.