Hello, I annotate my vcf with snpEff command like:

java -Xmx6g -jar snpEff.jar -c configfile GRCH38.86 > result.vcf

I checked ANN information at INFO field in result.vcf and found gene ND5 has 2 name( ND5 or MT-ND5 ) like:

C|upstream gene variant|MODIFIER|ND5|ND5|transcript|TRANSCRIPT ND5|protein coding||c.-4476T>C|||||4476|WARNING TRANSCRIPT MULTIPLE STOP COD

or

A|upstream gene variant|MODIFIER|MT-ND5|ENSG00000198786|transcript|ENST00000361567.2|protein coding||c.-4484G>A||

Why is that? Do I need to keep only one gene name?

It's likely due to alternative transcripts of the gene, since as you said, both of these are the same gene. I don't know all the annotation sources that snpEff uses, but the differences in the gene name/ID could be due to that. As to whether to keep it or not, that depends on if you're interested in the impact on potential isoforms or non-canonical transcripts. Is there any particular reason you don't want to just keep both of them?