UNCID_2668056.27d4c881-e906-437f-921d-467eb107798a.150108_UNC15-SN850_0402_AC5FTHACXX_2_GTTTCG.tar.gz
UNCID_2206882.58e94be5-2da2-4a96-944a-060f20216572.150108_UNC15-SN850_0402_AC5FTHACXX_3_CTTGTA.tar.gz
UNCID_2205121.2b3d596a-68ca-4c60-bb16-8c0cbd8fb190.140828_UNC11-SN627_0378_AC5FAWACXX_4_TAGCTT.tar.gz

xargs -n2 mv -i <<<"$(sed '1d; s/$/.tar.gz/' names.txt)"

Tar.gz_filenames       Sample_Names
UNCID_2668056.27d4c881-e906-437f-921d-467eb107798a.150108_UNC15-SN850_0402_AC5FTHACXX_2_GTTTCG.tar.gz                Sample2
UNCID_2206882.58e94be5-2da2-4a96-944a-060f20216572.150108_UNC15-SN850_0402_AC5FTHACXX_3_CTTGTA.tar.gz                Sample1
UNCID_2205121.2b3d596a-68ca-4c60-bb16-8c0cbd8fb190.140828_UNC11-SN627_0378_AC5FAWACXX_4_TAGCTT.tar.gz                Sample3

Sample1.tar.gz
Sample2.tar.gz
Sample3.tar.gz

150108_UNC15-SN850_0402_AC5FTHACXX_GTTTCG_L002_1.fastq
150108_UNC15-SN850_0402_AC5FTHACXX_GTTTCG_L002_2.fastq
150108_UNC15-SN850_0402_AC5FTHACXX_CTTGTA_L003_1.fastq
150108_UNC15-SN850_0402_AC5FTHACXX_CTTGTA_L003_2.fastq
140828_UNC11-SN627_0378_AC5FAWACXX_TAGCTT_L004_2.fastq
140828_UNC11-SN627_0378_AC5FAWACXX_TAGCTT_L004_1.fastq

Sample1_1.fastq
Sample1_2.fastq
Sample2_1.fastq
Sample2_2.fastq
Sample3_1.fastq
Sample3_2.fastq

tar xvzf *tar.gz

$ perl -lne '
    /.+?..+?.(S+)_(d+)_([^.]*)..*s+(.*)/;
    $n="$1_$3_L00$2";
    print "${n}_1t$4_1n${n}_2t$4_2"
' names.txt
150108_UNC15-SN850_0402_AC5FTHACXX_GTTTCG_L002_1    Sample2_1
150108_UNC15-SN850_0402_AC5FTHACXX_GTTTCG_L002_2    Sample2_2
150108_UNC15-SN850_0402_AC5FTHACXX_CTTGTA_L003_1    Sample1_1
150108_UNC15-SN850_0402_AC5FTHACXX_CTTGTA_L003_2    Sample1_2
140828_UNC11-SN627_0378_AC5FAWACXX_TAGCTT_L004_1    Sample3_1
140828_UNC11-SN627_0378_AC5FAWACXX_TAGCTT_L004_2    Sample3_2

while read old new; do mv -i "$old.fastq" "$new.fastq"; done < new.names.txt

tar xvzf *tar.gz &&
    perl -lne '
        /.+?..+?.(S+)_(d+)_([^.]*)..*s+(.*)/;
        $n="$1_$3_L00$2";
        print "${n}_1t$4_1n${n}_2t$4_2"
    ' names.txt |
        while read old new; do
            mv -i "$old.fastq" "$new.fastq";
        done