The Electors had total say, whoever was chosen in the curia was considered the rightful ruler.

Of course, this election process wasn't anything like a modern election. Ever since the Golden Bull of 1356 had standardised the election and crowning process you had seven (in the last century up to nine) fixed prince electors, who had been granted special rights like immunity from the Imperial court of law (privilegium de non evocando) and indivisibility of their lands (primogeniture). When the Emperor died a new one was to be crowned within 30 days, although in practice that not always appened of course.

The curia itself was made up of three ecclesiastic and four secular princes, fixed by the Golden Bull. The first three being the prince-bishops of Mainz, Cologne and Trier (who also held the positions of Archchancellor for Germany, Italy and Burgundy respectively) and the latter fours the King of Bohemia, the Count Palatine of the Palatine of the Rhine, the Duke of Saxony and the Margrave of Brandenburg.

During the Thirty Years War Catholic Bavaria received the Electorate status of the Protestant Palatinate and although the Palatinate regained its own status Bavaria was granted an additional electorial position in the Peace of Westphalia in 1648. Then in 1692 Emperor Leopold I. elevated the Duke of Braunschweig-Lüneburg (or Hannover) to be the ninth elector, in return the Duke promised his line's everlasting vote for the first-born Habsburg son. You might also remember Hannover as being in personal union with Britain from 1715 to 1837.

Austria itself was not an elector but held the same special powers as Archduchy. This dates back to a document forged in 1358 at the request of Rudolf IV., who was annoyed at not being recognised in the Golden Bull, now generally referred to as the Privilegium Maius - a reference to the Privilegium Minus of 1156, with which the then-Emperor elevated the March of Ostarrîchi to an independent Duchy, not the least because the forgers used the Imperial sigil from it and then destroyed the document.
Although the Privilegium Maius was so obviously forged (even including a signature by Julius Caesar) that the Emperor never even thought about recognising it, it was approved in 1452 when Frederick III regained the crown for the House of Habsburg.

Theoretically, the Electors had complete power over who to elect, as demonstrated in 1519 when they elected Charles V., the francophone King of Spain. Well, the only other one interested was the King of France but still...

In practice, the election of the Emperor's next of kin was an established practice long before the Habsburgs seized the position permanently. The Staufer dynasty held the title from 1138 to 1254 (with a brief 6-year interruption by the Welfen Otto IV. from 1209–1215), the Luxembourgs from 1312 to 1437 (with a few anti-Emperors).

As long as you kept the electors happy you could have reasonable expectation that you could reasonably expect that your heir had good chances of becoming your successor. But because you were Emperor and because you knew exactly who the guys with the powers were you weren't limited to hoping, it was a regular thing for the ruler to promote his son to the electors so they would chose him before the necessity arose. The heir to the Holy Roman Throne would then carry the honorary title King of Rome.
And because this was politics bribes were a must. Monetary sometimes, but most importantly the Wahlkapitulation (election surrender) or capitulatio caesarea. Ever since the aforementioned Charles V. it had become common that each candidate for the throne would provide a document detailing his responsibilities and concessions to the electors and empire in case he was chosen. As an illustration, here is the original Wahlkapitulation of Leopold I. from 1659 from the digital collection of the University of Mannheim.

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Now if you're asking about the Habsburg dynasty in particularly, well they were very good with playing the politics game. Ever since 1526 the Habsburgs had held the Crown of Bohemia, which granted them influence and a direct vote in the electorate curia and since 1570 the Crown of Hungary and Croatia with all the political, financial and military power that came with it. And as Archdukes the House of Habsburg, despite being a secular principality, also led the ecclesiastical bench of the Reichsfürstenrat in the Imperial Diet with even more opportunities of asserting influence.

And their large powerbase was also quite important, since the Empire was made up of so many tiny and splintered nations a ruler from those could offer neither the means to secure his election nor to ensure the safety of its members afterwards.