Two prominent advocates of this view are Rudolf Stier (1800–62) and Lloyd John Ogilvie (1930–). They argue both that the apostles erred, and that ultimately the evidence points to Paul being the true 12th apostle. They make several arguments:
The method of selection, particularly the casting of lots, was improper
The selection was unnecessarily hasty
There are a few things I would add to Nathaniel's answer.
There are additional points which may be drawn from the text.
As noted the selection of Matthias came before the outpouring of the Holy Spirit. It also came after Jesus had given commands through the Holy Spirit to the apostles whom He had chosen:
In the first book, O Theophilus,...
Reason for the choice of a new 12th Apostle
I agree with David Stratton, except I believe Peter gives the answer to your question in verses 20-22
Peter continued, “This was written in the book of Psalms, where it says, ‘Let his home become desolate, with no one living in it (Psalm 69:25).’ It also says, ‘Let someone else take his position (Psalm 109:8).’
Acts 1:8 Jesus orders the disciples to go to Jerusalem and await the coming of the Holy Spirit.
Acts 1:13 One hundred twenty disciples gather in an upper room at Jerusalem.
Acts 1:21 Peter determines basic qualifications for candidates for the 12th apostle.
No precedence for his qualification. Certainly not Jesus' method.
Many of the ...