Why did God choose tongues as the sign that a person has been filled with His Spirit? 

Perhaps the first reason, is that it is a fulfillment of Bible prophecy. 

"For with stammering lips and another tongue will he speak to this people. To whom he said, This is the rest wherewith ye may cause the weary to rest; and this is the refreshing: yet they would not hear." - Isaiah 28:11-12 

Paul refers to this scripture in 1 Corinthians: 

"In the law it is written, With men of other tongues and other lips will I speak unto this people; and yet for all that will they not hear me, saith the Lord. Wherefore tongues are for a sign, not to them that believe, but to them that believe not: but prophesying serveth not for them that believe not, but for them which believe." - 1 Corinthians 14:21-22  

{Note: “Sign” (“as tokens of Divine authority and power”) “Prophesying” (“signifies "the speaking forth of the mind and counsel of God"”) From: Vine’s Expository Dictionary of New Testament Words} 

"But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth." - Acts 1:8  

The disciples that were gathered in the upper room on the day of Pentecost were filled with the power of the Holy Ghost and spoke with tongues. The tongues was a sign they had been filled with divine authority and power. 

"And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance." - Acts 2:4  

Then, Peter stood up and prophesied, speaking forth the gospel under the anointing of the Holy Ghost to the Jews that had gathered.  

"Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." - Acts 2:38 

The second part of Isaiah’s prophecy of the Holy Ghost and tongues (Isaiah 28:12) speaks of the rest and refreshing of God’s Spirit. 

Jesus spoke of this rest, saying it comes from him. When we take upon us his yoke through baptism in Jesus name, he promises us his Spirit, which is our rest and refreshing. 

"Come unto me, all ye that labour and are heavy laden, and I will give you rest. Take my yoke upon you, and learn of me; for I am meek and lowly in heart: and ye shall find rest unto your souls. For my yoke is easy, and my burden is light." - Matthew 11:28-30  

"Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord;" - Acts 3:19

To be continued…