After watching this video about the limitations of democratic voting systems I spent some thoughts on the matter.
I feel like I came up with a pretty good systems in my head. The main idea is a ranked voting system that allows you to give additional weights to your choices. The counting happens recursively, unlike STAR voting which uses an arbitrary two step process. It can be summarized as a continuously ranked voting system.
Note

The limitations that are talked about in the video cannot be overcome mathematically, but some things are very unlikely to happen in the real world. Thus we can relax some of the properties that are used in Arrow’s impossibility theorem. 
Let’s make some stuff clear.
Note

The problem we are talking about
There are 
 Unanimity

Consensus is the ultimate goal.
 Anonymity

Equal voting power for each voter.
 Unrestricted Domain

Any voting will deterministically result in a decision.
 Transitivity

A > B
andB > C
implyA > C
. Every voter is able to decide on a ranking satisfying a total order.
 Independence of irrelevant alternatives

One of the main flaws of FPTP voting is that additional options can influence the result even though that option doesn’t even win. This is not desired considering transitivity.
 System abuse to increase voting power

Many ranked voting systems can be abused in a way by only voting for the single most preferred option. A well designed ranked voting system incentivizes to actually give the desired score to each option.
My idea of continuously ranked voting
Each voter will decide on an ordering of the options he wants to vote for. The voter will then distribute 100 points in a way that represents the ordering he has in mind. Giving the same number of points to multiple options is allowed.

10 points

10 points

0 points

30 points

50 points
Tip

Choosing the amount of exactly 100 points is very arbitrary. It’s just important to normalize the sum of points on every ballot to the same. The idea is to give the voter the ability to express something along these lines
I like option 
Counting continuously ranked ballots
We will eliminate options one by one. The worst option gets eliminated.
Let’s look at an example voting.
Options  A  B  C  D  E 

Ballot 1 
10 
10 
0 
30 
50 
Ballot 2 
0 
0 
100 
0 
0 
Ballot 3 
30 
30 
30 
0 
10 
Ballot 4 
50 
50 
0 
0 
0 
Ballot 5 
10 
20 
10 
0 
60 
Sum 
100 
110 
140 
30 
120 
In this example option D
will be eliminated first.
Ballot 1 was the only one that voted for that option.
To keep the voting power of Ballot 1 the same as before the 30 points that were given to option D
will now be distributed accordingly.
Options  A  B  C  E 

Ballot 1 
14.286 
14.286 
0 
71.428 
Ballot 2 
0 
0 
100 
0 
Ballot 3 
30 
30 
30 
10 
Ballot 4 
50 
50 
0 
0 
Ballot 5 
10 
20 
10 
60 
Sum 
104.286 
114.286 
140 
141.428 
Now we eliminate option A
and distribute the points the same way as before.
You might have noticed the recursion happening now.
Options  B  C  E 

Ballot 1 
16.667 
0 
83.333 
Ballot 2 
0 
100 
0 
Ballot 3 
42.857 
42.857 
14.286 
Ballot 4 
100 
0 
0 
Ballot 5 
22.222 
11.111 
66.667 
Sum 
181.746 
153.968 
164.286 
We keep going. We have arrived in the final round.
Options  B  E 

Ballot 1 
16.667 
83.333 
Ballot 2 
0 
0 
Ballot 3 
75 
25 
Ballot 4 
100 
0 
Ballot 5 
25 
75 
Sum 
216.667 
183.333 
In the final round option B
wins over option E
.
Pros and Cons of this voting method
The main idea of this counting method is to make sure that irrelevant alternatives really don’t change the outcome. It’s easy to see that in each recursion step no voting power is lost. Thus it is evident, that choosing irrelevant options is not harmful.
One problem is that ties can happen at any step. Even when the tie happens between irrelevant alternatives, you cannot just flip a coin and keep going. These ties can actually still affect the outcome of the overall vote. Luckily ties are quite unlikely. One could argue that this voting system doesn’t work on the "Unrestricted Domain" though.
Depending on the implementation, floating point rounding errors might affect the result.
Unfortunately it is not evident how your personal opinion translates to accurately giving points to options.
What I really like about this voting method though, is that it greatly incentivizes the use of your actual preference in your vote.
If you can spot a flaw with this system please let me know!
Also while writing this I stumbled upon rated voting. So I guess one could argue that this is just one way to do rated voting. Maybe it even exists already? Maybe "range voting" is very close?