While calling them literally Basques is a bit of an overstretch, the Jiménez were definitely of vasconian origin. This tribes, living isolated on the mountains, had enjoyed a great deal of autonomy from the romans (some even argue that they were de facto independent), and maintained it during the existence of the Visigothic Kingdom and up until the Umayyad conquest. Around the late 8th Century and early 9th, Carolingian influence crossed the Pyrenees and, in the form of Aquitanian and Gasconian lords, who were ik fact related to this Basques, established their dominance over the northernmost valleys of the area.

Charlemagne sacked Pamplona during his campaign against Cordoba and, in revenge, the basques, lead by one Íñigo Arista, ambushed the rear of the frankish army as it was crossing the passes of Roncesvaux and anahilated it, killing the famous knight Roland as the epic poem relates. It is around these times when the sources speak of a certain Jimeno the Strong, who, if real, was the ancestor of the Jiménez line.

After this, and specially after defeating, again in Roncesvaux a few decades later, another frankish army, the successors of Arista managed to create a semi-independent realm that was soon to be taken over and ruled by the Jiménez, who where probably related to Arista's own dynasty. During a hundred years, the kings of Pamplona slowly pushed down from the mountains, taking advantage from muslim internal struggles, and managed to create a small but strong kingdom that even annexed some of the weaker frankish counties to its east, who had become de facto autonomous since the Carolingians started their self destruction after the death of Louis the Pious.

In the early 11th Century, a young, able and energetic Sancho III ascended to the throne of Pamplona and, with a weak Kindgom of León to his west, and an unstable and soon-to-be-ripped-appart-by-its-nobility Caliphate of Córdoba, gained control of the County of Castile, which was essentially the eastern half of the Kingdom of Leon, and married his younger son to the sister of young king Bermudo III of León, becoming the most powerful ruler of the Iberian Peninsula. After his sudden death in 1035, his older legitimate son, García III, took the crown of Navarra, while his other son, Fernando, was given Castille, and his illegitimate son, Ramiro, was granted the County of Aragón, to be ruled as a king, but under the supremacy of Pamplona.

Fernando would go on and kill his brother in law in battle in 1037, taking over León by virtue of his wife's claim, and would rule there until 1065, when his kingdom was divided between his sons, Sancho, ruler of Castile, Alfonso, king of León, and García, ruler of Galicia. Alfonso would eventually manage to reunite the three kingdoms and ruled over them until 1109 García, on the other hand, was also killed in battle by Fernando in 1054, leaving behind his infant son, Sancho IV, as king of Pamplona. Ramiro would be succeded bu his son, Sancho Ramírez, after his death in 1063. (I know, a ton of Garcías and Sanchos, it is a bit of a head pain for me as well)

Now, while these three would've been pretty much as basque as their father, not the same can be said about their own sons. Sancho IV, on the one hand, would've been educated by basque courtiers, but on the other hand, a differenr thing was going on with his cousins, since Alfonso grew up surrounded by the sons of the most importan leonese and castilian nobility, and Sancho of Aragón most certainly did the same.

In León, Alfonso VI was succeded by his daughter Urraca, who, by the time, was only basque as in having a basque name. Even more, any potential basque identity, and even the Jimena dynasty itself, disappeared when his son by Raymond of Burgundy, a french nobleman who Alfonso VI had given the county of Galicia as dowry when he married Urraca, Alfonso VII, succeded her in 1126, inaugurating the Burgundian house of León and Castile.

In Aragón, Sancho would be succeded by his son, Pedro I, who was in turn succeded by his brother, Alfonso I. This Alfonso died also without issue and the third brother, Ramiro, was proclaimed king. Ramiro, however, had lived a religious life since his youth, as he was never expected to inherit (even though he did had the vocation), was not married, and had little to no interest in becoming king. And so, he married a french lady, had a daughter, arranged the bethrothal of this daughter, Petronila, to the already adult count Ramón Berenguer IV of Barcelona, abdicated and, just like that, went to live the rest of his days in the same monastery he was in when the nobility recalled him to succeed his brother. When Petronila became of age, she married Ramón, by whom he had a son, Alfonso, who succeded them both as the first ruler of the Crown of Aragón, being both king of Aragón and count of Barcelona. And, also, not a member of the Jimena dynasty, that became extint on the aragonian branch when Petronila died.

Pamplona, what would later be called Navarra, would lose its independence when Sancho IV was murdered and Sancho Ramírez of Aragón was crowned as his successor, but after the death of Sancho's son and eventual heir, Alfonso I, the basque nobility chose as their king the grandson of an illegitimate son of king García III, and so the basque line was restored in the figure of García Ramírez, who is actually known as the Restorer, and would continue on his son, Sancho VI, and his grandson, Sancho VII. This chap, who was, according to sources, over 2 metres tall and rode a mule to battle since no horse could resist his size, would participate in the battle of the Navas de Tolosa in 1212. However, he died with no legitimate issue some time after, in 1234. This marked officially the end of the dynasty, since the basque nobility offered the crown to Theobald IV, count of Champagne, and son of Sancho's sister, Blanche. This marked the start of two long centuries of political complications in Navarre, which eventually passed to the Capets themselves, and then to a cadet branch after they were replaced by the house of Valois, only to be fought over in a complex dynastic strife between Castile, Aragón and France until, in 1514, Ferdinand II of Aragón definitely conquered Navarra and annexed it to the Crown of Castile, to which it would be attached until its abolition as a kingdom in 1841, result of its support in a civil war to a claimant to the Spanish throne.

So, all in all, and while the dynasy of Jimeno the Strong gave some of the most powerful and important kings of Medieval Spain, such as Fernando I and Alfonso VI of León or Alfonso I of Aragón, its legacy was pretty much a short lived one, specially outside of its ancestral home of Pamplona.