If a rogue gains a second action during a surprise round (through Haste, action surges, etc.), can he ready the second action to interrupt an enemy's turn while they are surprised? Or does there need to be an action to key off of?
There is no surprise round in D&D 5, there is only a surprised creature!
Therefore, I don't think your question makes sense.
Assuming the hasted rogue (HR) is not surprised and the surprised victim (SV) is, the first thing that happens is that they each roll for initiative. There are 2 possible outcomes:
SV beats HR SV takes their turn, being surprised they can do nothing but they are now no longer surprised. HR then takes their turn which can include as many Ready actions as they have actions available based on a perceivable trigger.
HR beats SV HR takes their turn while SV is surprised. This can include as many Ready actions as they have actions available based on a perceivable trigger. Given that SV is going to do nothing on their turn it is difficult to see what the trigger could be. After their turn SV can take a reaction and this could serve as a trigger provided there is someone else in the combat to cause SV to take a reaction. If it is just the 2 of them then it is HR's turn and the Readied actions are wasted. I can't see why you just wouldn't use the extra actions to e.g. attack SV while they are still surprised i.e. on HR's turn.
This will depend on your DM
According to the text for a readied action:
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn so that you can act later in the round using your reaction. First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.
So, you can ready an action during a surprise round, however in order to use it you'll need to give your DM the perceivable circumstance that you're interested in. Some DMs may consider 'the enemy starts their turn' as perceivable, but some may not. It is definitely a 'meta' approach (turns aren't real things, they're abstractions), and many DMs would not let you do that.
Also, consider the fact that the text later says:
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
So whatever trigger you set has to occur first, then you can act. So if you say something like 'the enemy begins their turn', and your DM accepts that, the DM is well within their rights to determine what the 'beginning' of the turn includes. That may allow the target to attack back, slink away or even take the dodge action at the beginning of their turn, which would make it harder for you to hit your action (or even potentially lose your action altogether).