Begin by considering what you are engineering: You need to create a belief, you need to create it in enough people and they need to be the right people. You can put unambiguous evidence in front of some people and they will still refuse to believe it ( consider any of the current anti-scientific movements ) but most of those people are not the same ones who are in the upper echelons of power. Those need to be your first audience, they will carry more of the people with them.
Rational people tend to build up beliefs through a steady build up of evidence and your character's endeavour may be more persuasive if rather than a single event they had already warmed people up to the idea and created a narrative around it- our brains are irresistibly drawn towards stories, so this could work well.
So we might start by asking what aliens would do that would result in a first contact event. That also needs us to consider what or who the aliens are.
Our budget probably doesn't stretch to a full-on landing, so perhaps the aliens are just passing through ( Rama style ) or investigating Earth in some way, either in a way that would incorporate direct contact with humans or not. Alternatively they may be doing something entirely inscrutable, which requires some very original thinking before one can suitably design it. There is a lot to be said for the latter case as we can create a series of unconnected and unexplained events, which people will draw their own conclusions from and build up connections.
Elements that could be useful:
- Satellites capable of transmitting radio signals - if you want interactivity then these might be your messengers, if not then they could be broadcasting "communications" with unknown entities on the planet.
- A technology that allows you to do much of what can be done electronically without electronics. This is a tricky one as you either need something mechanical- for example circuitry based on magnetic rather than electrical induction, entirely analogue computation or something that behaves like a biological system, using proteins for data storage, communication and so on. You might find that evolvable hardware might help create weird designs. The thing about implementing a technology like this is that to get it sufficiently advanced to be credible for a space-faring species you probably need to get it very sophisticated and then the temptation to make yourself even richer with a vast portfolio of crazy patents would be difficult to resist, although this could be part of your post-event plan where you could claim to be analysing it. Anything of this kind must be carefully synthesized in a way that favours materials that do not occur on earth. This is your representative among humans so it must be very detailed.
- Communications that are totally weird. Are they meaningful or not? Ideally you might want to go for something reminiscent of the Voynich Manuscript which appears to be meaningful but may not be. For extra points avoid binary logic- encoding it through an analogue process or using a logic that operates in an unusual base.
- A way of ensuring your technology gets to do its thing noticeably. If you have your own island then a good way of deploying it would be from a couple of stealthed-up submarines. In fact if we were to see the visitors as aquatic then they might not even be interested in the surface and the drones above would be more like fish. The oceans are vast and little explored, so an aquatic alien would be hard to find if they were smart and unwilling to be found. Ensuring a few drones turn up in fishing nets would start to draw attention to them.
With these in place, you might choose to start by waiting until a noticeably large meteor hits the sea- this happens a lot but for extra points you might just begin the endeavour during one of the big meteor showers. Pick a starting point in the sea and release a big old burst of your weirdly encoded transmission. Have the satellites mirror it very briefly ( perhaps for maximum effect the satellites are positioned so that they can "animate" a moving signal source across the sky.
After a few months of silence, another burst of encoded data from another location. People may be listening for it by now, and finding it will certainly draw them in.
Then, the first drone is found in a fishing net somewhere. For maximum effect you might give it a form and swimming mechanism that is entirely different from any terrestrial creature.
A few more drones turn up - they need to appear in different places at similar times, to create a clear impression of worldwide spread. Some of them will be getting analysed and questions will be asked everywhere, but it is inevitable that some will be encountered by regular people and videos will be out online fairly quickly. A few different varieties of drone would help to create interest.
From time to time drones emit a short burst of inscrutable data. Once they are out of the water ( or out of salt water ) this takes on a standard pattern.
Once everyone is talking about this mystery, it's time for your grand finale. Now I don't have a clear picture of what this should be, but follow it up by a sequential and dramatic destruction of the signal satellites, designed to create the impression of something powering itself around the planet and away in a series of explosive bursts. Ideally all drones "die" at this point. The aliens have gone, you just have to clean up all evidence of their creation.
The biggest risk is that the more people are involved in a conspiracy the sooner it fails, so you need to have a very dedicated pranking crew behind you and you can't expect the story to hold forever. You almost certainly can however, with the right resources, persuade major powers of extra-terrestrial visits at least temporarily and create a mystery that will fascinate across generations.