​ Good question. Your focus both on "mestizos" and the 17th century makes it more difficult to answer, because most information we have is on native people in the 16th century. I'll talk a bit about why that is the case and then in a second step look into the few examples I can think of regarding mestizos.

I.

The first development here concerns the enslavement of native people in Spanish America. Nancy van Deusen has written a great book ("Global Indios") on this, where she describes distinct phases: First between 1500-1542 "the enslavement of hundreds of thousands of people from America and elsewhere" (including Africa) due to the "open-ended exceptions of just war and ransom". Just war had served as a justification for war against muslims in medieval Iberia and continued to be used for conquest campaigns in the Americas. A second phase begins with the New Laws of 1542 under Charles and heavily influenced by Bartolomé de las Casas. These stated that native Americans were human, vassals of the Spanish Crown and free - effectively prohibiting enslavement of native people for just war or ransom.

However, the New Laws included important loopholes which led to enslavement of natives continuing circa until the late 16th/early 17th century, albeit in much smaller numbers (numbering rather in the thousands regarding Castile). This meant that native people from Spanish America were still being brought to Spain at that time, often via Portugal. They would then use legal mechanisms open to them to argue for freedom, often succesfully.

You'll notice I've been talking about native people here, called "indios" by the Spaniards, rather than mestizos. This was to show that native people were actually brought to Europe in large numbers in the 16th century, but much less so in the 17th century.

Regarding mestizos, one thing to keep in mind is that this concept came into more regular use only in the late 16th century, not before. The other thing is that "mestizos" formed part of a social system in Spanish America known as the casta system. This was a very complex system, and I'll keep it brief here so to stick with your question.

Basically the casta system distinguished between "main groups", especially Spaniards, criollos (Europeans born in Americas), indios and Africans, and then various subgroups including mestizos (descending from natives and Europeans as you say). While the Spanish were at the top of the hierarchy, the natives also held a special position because the Spanish justification for colonising the Americas basically rested on the voluntary conversion to CHristianity of its population. This also meant that (at least in the 16th c.) traditional native elites e.g. in Mexico or Peru still held some more important political and economic powers than a "mixed" group like the mestizos.

However, it's also important to keep in mind that at his time the boundaries between the groups were not yet as fixed as in later centuries; so that we have many cases of people "changing" between casta groups when it was of advantage. And also that by the turn of the 17th century, in places like Mexico the native elites had lost much influence to Spaniards, mestizos and other groups.

Lastly, this is important for your questions because (again, esp. in the 16th century) a mestizo person was not necessarily recognised as "mestizo" with the category coming into use only then; s/he would have probably belonged to a lower class of society, compared to e.g. the native nobility; and it would have thus been quite improbable for a mestizo person to make it to Europe, except for the exceptions who had made it into the higher classes somehow.

While there would have been more "elite mestizos" in the 17th century, I also guess there would not have been a great incentive to leave their high status/properties behind, which often took generations to attain, in order to leave for another continent. What is more, "mestizo" afaik did not figure in board books as an ethnic category (like "indio", African etc.), so it would be much harder to actually know about these cases - especially as former slaves in Spanish courts tried to position themselves as "indio" because of the natives' special rights mentioned above.

II.

So we've seen that enslaved native people were brought to Iberia until ca. the 17th century. And that there was little probability of lower class mestizos coming to Europe. This leaves us with elite native and (esp. for your question) elite mestizo people. While there were cases of delegations of native elites coming to Spain, I've only seen very few cases of them actually moving there. However, as my main focus lies on colonial Mexico it might there are cases e.g. from South America that I haven't come across.

As for such delegations, occasionaly a mestizo noble would form part of the indigenous nobility traveling to Spain. One such case was with Diego de Muñoz Camargo, who travelled in 1585 to the court with his delegation from Tlaxcala (a region and city in central Mexico), and handed over one of his historical writings to king Philipp II. We can see this as a rather exceptional case, because the Muñoz Camargos had become wealthy through economical enterprises and through the younger Diego de Muñoz (his father had the same name) marrying a member of Tlaxcala's highest native elite. Incidentally the delegation was succesful in securing further special rights for Tlaxcala.

Anyway, I'll now talk about the two cases I know of that come closest to what you asked about: Those of Juan Cano de Moctezuma and, probably most famously of the Inca Garcilaso de la Vega. "Unfortunately" both moved to Spain already in the late 16th century as well.

The Aztec/Mexica ruler Moctezuma had many children, and from those that survived the war with the Spaniards three of his daughters and one son received important rights to native labor (or encomiendas). The largest of these encomiendas went to his daughter doña Isabel who married a Spaniard, so that officially their children would fit your description of being mestizo. I wrote some more about Moctezuma's descendants in this earlier post, and will quote from it on their son Juan Cano de Moctezuma:

I'll now turn to one branch of the extended Moctezuma family who would remain as nobility in Spain: The condes de Moctezuma. Unfortunately I haven't been able to find as much about them as about the Mexican branches - possibly a sign that less was written about them at the time. Juan Cano de Moctezuma, a son of donha Isabela and Juan Cano de Saveedra, traveled to Cáceres in the early 1550s and married Elvira de Toledo in 1559, which made his children counts and dukes - and him the most successful of donha Isabela's heirs. His mayorazgo in Granada was formalized by the Crown in 1577. At the same time, Juan Cano was involved in complicated lawsuits on the Mexican Tacuba encomienda of his mother with his siblings (detailed by Donald Chipman), and would continue to receive his share.

Jumping in time, we have one count of Moctezume receiving the post of viceroy for large sums of money, as was customary by the late 17th c. Accompanied by his two daughters and wife he sailed to Mexico City in 1696. According to Chipman, with his viceroyalty began a time of greatly reduced spending for viceroys, and he describes the conde's viceroyalty as "uneventful". So while we have a Spanish noble who descended from Moctezuma "returning" to Mexico to take up the highest post, this act doesn't appear very influential. A few words on the family's title: In 1865 the title of "duchy of Moctezuma" was created by queen Isabel II. in favor of the XIV. count of Moctezuma. In 1992, the title was extented as the "duchy of Moctezuma de Tultengo" - in reference to the Mexican city Tultengo, already featured in the name of the counts. The family hold the title to this day with many properties all over Spain, and there are also other noble families who trace their lineage back to Moctezuma. On the other hand, the descendants of donha Isabel would continue to recceive payments by the Mexican state until 1934, with later descendants apperently currently trying to receive payments once more.

So you could probably argue that Juan Cano's descendants would count as mestizos living in Spain in the 17th century. Then again, it seems to me the condes de Moctezuma received their title due to intermarriage with Spanish nobles, and were perceived as such than as native American or mestizo.

The other case is that of the famous chronicler 'Inca' Garcilaso de la Vega from Peru. His is an exceptional case as he descended both from the highest pre-conquest Inca elite (tied to the Inca) and from Spanish conquistadors. This enabled him to move to Spain with the support of an uncle in 1560, where he sought the acklowledgement of privileges from his father's side. His chronicle "Comentarios Reales de los Incas" proved to be a success in Europe, and de la Vega remained in Spain until his death.

As I said this is quite an exceptional case, as the other major Spanish American native and mestizo authors known to us from the 16th c. were not published in their life-time and remained in the Americas (except for occasional petitions in Madrid, see above). So it seems like no coincidence that the writer who actually managed to move to Spain due to his good family connections managed to get published, and even gain a sort of „prominence“. It probably also had to do with his writing over aspects of native Andean culture and esp. of faith that would have been considered offensive/non-Christian in Spain.