I like how the Westminster Larger Catechism answers this question. We used it in worship a few weeks ago (lightly edited and broken into lines for readability).
To pray in the name of Christ is,
in obedience to His command,
and in confidence in His promises,
to ask mercy for His sake;
not by simply mentioning His name,
but by drawing our encouragement to pray,
and our boldness, strength, and hope of acceptance in prayer,
from Christ and His mediation.