There are strict halachot about what can be done with the money earned from the sale of a holy article. The mishna[1] says that when one sells an article of kedusha, he must use the proceeds to buy something of higher kedusha. However, the gemara[2] says that if the city’s leadership sells the item in the presence of the populace, thereby obtaining maximum authority, the money can be used freely. The Tur[3] maintains that since every individual controls his own property, he can similarly sell it and use the money freely.
On the other hand, the gemara[4] asserts that one may sell a sefer Torah only to enable him to learn Torah or to get married. The Beit Yosef[5] infers from the fact that the gemara’s source and other sources are written in the singular that this directive is referring to a privately owned sefer Torah; nevertheless, it is permitted to sell it only under very specific circumstances. The Shulchan Aruch[6] cites two opinions regarding whether, under normal circumstances, it is permitted to sell a privately owned sefer Torah. (If a privately owned sefer was dedicated to community use, the matter is more severe). While there is no consensus in deciding between the opinions, many have been lenient.[7]
The question of selling privately owned tefillin is significantly simpler. Besides the fact that the sanctity of a sefer Torah is greater than that of all other religious articles, there is a special mitzva to write/possess one. Thus, if one sells his only sefer Torah, he may be uprooting the fulfillment of that mitzva, and it is usually forbidden to do so. However, if the sefer Torah was acquired in a manner in which one did not fulfill the mitzva, it is more likely to be permitted to sell it.[8] Likewise, if one has another sefer Torah and thus will continue to fulfill the mitzva even after the sale, there are greater grounds for selling it.[9] In any case, the Magen Avraham[10] says that the prohibition for an individual to sell does not apply to holy articles other than a sefer Torah, such as tefillin. While not everyone agrees,[11] the consensus seems to be on the lenient side.[12]
In your case, there are further reasons for leniency. Obviously, it cannot always be forbidden to sell tefillin for normal profit. If it were, what would a producer or merchant of religious articles do? The Beit Yosef[13] writes that they are permitted to sell because those holy articles were always slated for sale and not for their own personal use. The Kaf HaChayim[14] applies this logic to other cases, such as one who receives a sefer Torah as payment of a debt or as an inheritance. He reasons that the permissibility to sell depends on the owner’s intent when the article entered his possession. In the case of inheritance, we consider the inheritor’s intent upon receiving the sefer Torah. If he planned to use it, then the aforementioned questions arise about switching its designation and selling it. If when he received it he had in mind to sell it, he would not be changing its designation, and selling would therefore be permitted. Since you fit into the latter category, leniency is indicated again, and even more so since we are referring to tefillin rather than a sefer Torah.
Your financial situation can also be a significant factor. If your financial situation is very difficult, it is possible that you need the money so that you can continue learning Torah with the regularity that you desire, and the gemara states that this is justification for selling even a sefer Torah. If your situation is less dire and you have the merit of giving significant tzedaka, you may prefer not to sell your grandfather’s tefillin and instead donate them to someone in need. The money you save the recipient can be credited as tzedaka, thereby lowering your out-of-pocket tzedaka expenses.